Advanced Computer Techniques in Applied Electromagnetics (Studies in Applied Electromagnetics)

ADVANCED COMPUTER TECHNIQUES IN APPLIED ELECTROMAGNETICS Studies in Applied Electromagnetics and Mechanics Series Edit...

Author: Slawomir Wiak | Andrzej Krawczyk | Ivo Dolezel

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ADVANCED COMPUTER TECHNIQUES IN APPLIED ELECTROMAGNETICS

Studies in Applied Electromagnetics and Mechanics Series Editors: K. Miya, A.J. Moses, Y. Uchikawa, A. Bossavit, R. Collins, T. Honma, G.A. Maugin, F.C. Moon, G. Rubinacci, H. Troger and S.-A. Zhou

Volume 30 Previously published in this series: Vol. 29. Vol. 28. Vol. 27. Vol. 26. Vol. 25. Vol. 24. Vol. 23. Vol. 22. Vol. 21. Vol. 20. Vol. 19. Vol. 18. Vol. 17. Vol. 16. Vol. 15. Vol. 14. Vol. 13. Vol. 12.

A. Krawczyk, R. Kubacki, S. Wiak and C. Lemos Antunes (Eds.), Electromagnetic Field, Health and Environment – Proceedings of EHE’07 S. Takahashi and H. Kikuchi (Eds.), Electromagnetic Nondestructive Evaluation (X) A. Krawczyk, S. Wiak and X.M. Lopez-Fernandez (Eds.), Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering G. Dobmann (Ed.), Electromagnetic Nondestructive Evaluation (VII) L. Udpa and N. Bowler (Eds.), Electromagnetic Nondestructive Evaluation (IX) T. Sollier, D. Prémel and D. Lesselier (Eds.), Electromagnetic Nondestructive Evaluation (VIII) F. Kojima, T. Takagi, S.S. Udpa and J. Pávó (Eds.), Electromagnetic Nondestructive Evaluation (VI) A. Krawczyk and S. Wiak (Eds.), Electromagnetic Fields in Electrical Engineering J. Pávó, G. Vértesy, T. Takagi and S.S. Udpa (Eds.), Electromagnetic Nondestructive Evaluation (V) Z. Haznadar and Ž. Štih, Electromagnetic Fields, Waves and Numerical Methods J.S. Yang and G.A. Maugin (Eds.), Mechanics of Electromagnetic Materials and Structures P. Di Barba and A. Savini (Eds.), Non-Linear Electromagnetic Systems S.S. Udpa, T. Takagi, J. Pávó and R. Albanese (Eds.), Electromagnetic Nondestructive Evaluation (IV) H. Tsuboi and I. Vajda (Eds.), Applied Electromagnetics and Computational Technology II D. Lesselier and A. Razek (Eds.), Electromagnetic Nondestructive Evaluation (III) R. Albanese, G. Rubinacci, T. Takagi and S.S. Udpa (Eds.), Electromagnetic Nondestructive Evaluation (II) V. Kose and J. Sievert (Eds.), Non-Linear Electromagnetic Systems T. Takagi, J.R. Bowler and Y. Yoshida (Eds.), Electromagnetic Nondestructive Evaluation

Volumes 1–6 were published by Elsevier Science under the series title “Elsevier Studies in Applied Electromagnetics in Materials”.

ISSN 1383-7281

Advanced Computer Techniques in Applied Electromagnetics

Edited by

Sławomir Wiak Institute of Mechatronics and Information Systems, Technical University of Lodz, Poland

Andrzej Krawczyk Institute for Labour Protection, Warsaw, Poland

and

Ivo Dolezel Technical University of Prague, Czech Republic

Amsterdam • Berlin • Oxford • Tokyo • Washington, DC

© 2008 The authors and IOS Press. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without prior written permission from the publisher. ISBN 978-1-58603-895-3 Library of Congress Control Number: 2008931370 Publisher IOS Press Nieuwe Hemweg 6B 1013 BG Amsterdam Netherlands fax: +31 20 687 0019 e-mail: [email protected] Distributor in the UK and Ireland Gazelle Books Services Ltd. White Cross Mills Hightown Lancaster LA1 4XS United Kingdom fax: +44 1524 63232 e-mail: [email protected]

Distributor in the USA and Canada IOS Press, Inc. 4502 Rachael Manor Drive Fairfax, VA 22032 USA fax: +1 703 323 3668 e-mail: [email protected]

LEGAL NOTICE The publisher is not responsible for the use which might be made of the following information. PRINTED IN THE NETHERLANDS

Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved.

v

Preface This book contains papers presented at the International Symposium on Electromagnetic Fields in Electrical Engineering ISEF’07 which was held in Prague, the Czech Republic, on September 13–15, 2007. ISEF conferences have been organized since 1985 as a common initiative of Polish and European researchers who deal with electromagnetic field applied to electrical engineering. Until the present the conferences have been held every two years either in Poland or in one of European academic centres renowned for electromagnetic research. Technical University of Prague and the Chech Academy of Sciences make Prague be such a centre. Additionally, Prague is well-known in the world for its beauty and charm and it is called “Golden Prague”. The city of Prague is one of the six most frequently visited cities in Europe. Indeed, it is indisputable that Prague can attract every has the opportunity to visit it. The long, more then 20-year-old, tradition of ISEF meetings is that they try to tangle quite a vast area of computational and applied electromagnetics. Moreover, ISEF symposia aim at joining theory and practice, thus the majority of papers are deeply rooted in engineering problems and simultaneously present high theoretical level. Bearing this tradition, we attempt to touch the core of electromagnetic phenomena. After the selection process 237 papers were accepted for the presentation at the Symposium and almost all of them were presented at the conference, both orally and in the poster sessions. The papers have been divided into the following groups: • • • • • • •

Micro and Special Devices Electromagnetic Engineering Computational Electromagnetics Coupled Problems and Special Applications Measurement Monitoring and Testing Techniques Bioelectromagnetics Magnetic Material Modelling

The papers which were presented at the symposium had been reviewed and assessed by the sessions’ chairmen and the Editorial Board assembled for the postconference issue of ISEF’07. All the papers accepted for further publication were divided into three groups: 1) of more computational aspect, 2) of information technology aspect and 3) of more applicable nature. The latter ones are published in this volume while the first ones went to COMPEL journal (COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 27, No. 3/2008) and the second group to Springer Verlag (series on Studies in Computational Intelligence, vol. 119, 2008). The papers selected for this volume have been grouped in three chapters and seven sub-chapters. The division introduces some order in the pile of papers and the titles of chapters mirror the content of the papers to some extent. Names of chapters and subchapters are as follows: Chapter A Fundamental Problems and Methods Fundamental Problems Methods

vi

Chapter B Computer Methods in Applied Electromagnetism Computational Methods Numerical Modelling of Devices Chapter C Applications Electrical Machines and Transformers Actuators and Special Devices Special Applications The papers gathered in Chapter A are mainly devoted to physics of electromagnetic materials and mathematical approaches to electromagnetic problems. In the first sub-chapter papers concern physical phenomena, like magnetostriction, vibrations, anisotropy, occuring in the various electromagnetic materials from ferromagnetics to dielectromagnetics. And the second sub-chapter consists of papers concerning methods of analysis of electromagnetic phenomena in their methodological aspects. Chapter B contains papers dealing with numerical (or computer) analysis of electromagnetic devices and phenomena. The first sub-chapter shows how mathematical methods are realised numerically, i.e. how to make real calculation, based on numbers. And the papers gathered in the second sub-chapter deal with numerical modelling of some groups of devices. Chapter C, in turn, reveals the world of engineering problems, showing how theoretical and methodological considerations can be transferred to real engineering problems. Indeed, the chapter gives the image of real applied electromagnetics. The first sub-chapter is devoted to the very classical electrical devices, namely transformers and electrical machines. In spite of avery long tradition of numerical analysis of electromagnetic phenomena in such devices, the papers bring some new ideas and approaches. The second sub-chapter shows newer applications like sensors and actuators, and thus the area of engineering called mechatronics. Special approaches are needed inthe analysis of these devices as their size and operation features are quite different fromthe previous ones. And the last sub-chapter gathers a few papers dealing with very special applications based, for example, on superconductivity or ferroresonance. Needless to add that the electromagnetic analysis in such cases requires again new techniques and methods. The division of the papers is far from clear distinction of the papers’ topics and content. It is a very rough distinction which gives prospective readers some suggestion on how to find a paper of their personal interest. Summarising this introductory remarks we, the Editors of the book, would like to express our hope that the book you have in your hands will help the world-wide electromagnetic community, both academic and engineering, in better understanding electromagnetism itself and its application to technical problems. At the end of these remarks let us be allowed to express our thanks to our colleagues who have contributed to the book by submitting their papers or/and by peerreviewing the papers at the conference as well as in the publishing process. We also convey our thanks to IOS Press Publisher for their effective collaboration in giving this very attractive shape of the book and its promoting. Let us also express our strong belief that ISEF conference will maintain strong links with IOS Press in the future. Ivo Dolezel Chairman of the Organising Committee

Andrzej Krawczyk Scientific Secretary

Sławomir Wiak Chairman of the ISEF Symposium

vii

Contents Preface Ivo Dolezel, Andrzej Krawczyk and Sławomir Wiak

v

Chapter A. Fundamental Problems and Methods A1. Fundamental Problems Power Effect in Magnetic Lamination Taking into Account Elliptical Hysteresis Approach Kazimierz Zakrzewski

3

Study of Electromagnetic Field Properties in the Neighbourhood of the Metallic Corners Stanisław Apanasewicz and Stanisław Pawłowski

8

Eddy Currents and Lorentz Forces in Pulsed Magnetic Forming Jan Albert, Wolfgang Hafla, André Buchau and Wolfgang M. Rucker Applicability of Microwave Frequencies to the Evaluation of Deeper Defects in Metals Dagmar Faktorová

16

21

Electromagnetic Field Energy in Ferromagnetic Barriers Ryszard Niedbała, Daniel Kucharski and Marcin Wesołowski

26

The Influence of Temperature on Mechanical Properties of Dielectromagnetics Barbara Slusarek, Piotr Gawrys and Marek Przybylski

34

Influence of the Magnetic Anisotropy on Electrical Machines M. Herranz Gracia and K. Hameyer

39

Analysis of Structural Deformation and Vibration of Electrical Steel Sheet by Using Magnetic Property of Magnetostriction Wataru Kitagawa, Koji Fujiwara, Yoshiyuki Ishihara and Toshiyuki Todaka

47

A2. Methods Rapid Design of Sandwich Windings Transformers for Stray Loss Reduction J. Turowski, Xose M. Lopez-Fernandez, A. Soto Rodriguez and D. Souto Revenga

53

Application of Logarithmic Potential to Electromagnetic Field Calculation in Convex Bars Stanisław Apanasewicz

58

Multi-Frequency Sensitivity Analysis of 3D Models Utilizing Impedance Boundary Condition with Scalar Magnetic Potential Konstanty Marek Gawrylczyk and Piotr Putek

64

viii

Very Fast and Easy to Compute Analytical Model of the Magnetic Field in Induction Machines with Distributed Windings Manuel Pineda, Jose Roger Folch, Juan Perez and Ruben Puche Coupling Thermal Radiation to an Inductive Heating Computation Christian Scheiblich, Karsten Frenner, Wolfgang Hafla and Wolfgang M. Rucker Consideration of Coupling Between Electromagnetic and Thermal Fields in Electrodynamic Computation of Heavy-Current Electric Equipment Karol Bednarek Force Computation with the Integral Equation Method Wolfgang Hafla, André Buchau and Wolfgang M. Rucker

72 80

85 93

Chapter B. Computer Methods in Applied Electromagnetism B1. Computation Methods Numerical Simulation of Non-Linear Electromagnet Coupled with Circuit to Rise up the Coil Current Slawomir Stepien, Grzegorz Szymanski and Kay Hameyer

101

Numerical Calculation of Power Losses and Short-Circuit Forces in Isolated-Phase Generator Busbar Dalibor Gorenc and Ivica Marusic

108

Numerical Methods for Calculation of Eddy Current Losses in Permanent Magnets of Synchronous Machines Lj. Petrovic, A. Binder, Cs. Deak, D. Irimie, K. Reichert and C. Purcarea

116

3-D Finite Element Analysis of Interior Permanent Magnet Motors with Stepwise Skewed Rotor Yoshihiro Kawase, Tadashi Yamaguchi, Hidetomo Shiota, Kazuo Ida and Akio Yamagiwa Advance Computer Techniques in Modelling of High-Speed Induction Motor Maria Dems and Krzysztof Komęza

124

130

Computation of the Equivalent Characteristics of Anisotropic Laminated Magnetic Cores E. Napieralska-Juszczak, D. Roger, S. Duchesne and J.-Ph. Lecointe

137

Improving Solution Time in Obtaining 3D Electric Fields Emanated from High Voltage Power Lines Carlos Lemos Antunes, José Cecílio and Hugo Valente

144

Thermal Distribution Evaluation Directly from the Electromagnetic Field Finite Elements Analysis A. di Napoli, A. Lidozzi, V. Serrao and L. Solero

151

Coordination of Surge Protective Devices Using “Spice” Student Version Carlos Antonio França Sartori, Otávio Luís de Oliveira and José Roberto Cardoso

158

ix

B2. Numerical Models of Devices Nonlinear Electromagnetic Transient Analysis of Special Transformers Marija Cundeva-Blajer, Snezana Cundeva and Ljupco Arsov

167

Some Methods to Evaluate Leakage Inductances of a Claw Pole Machine Y. Tamto, A. Foggia, J.-C. Mipo and L. Kobylanski

175

Reduction of Cogging Torque in Permanent Magnet Motors Combining Rotor Design Techniques Andrej Černigoj, Lovrenc Gašparin and Rastko Fišer

179

Optimum Design of Linear Motor for Weight Reduction Using Response Surface Methodology Do-Kwan Hong, Byung-Chul Woo, and Do-Hyun Kang

184

Analytical Evaluation of Flux-Linkages and Electromotive Forces in Synchronous Machines Considering Slotting, Saliency and Saturation Effects 192 Antonino di Gerlando, Gianmaria Foglia and Roberto Perini Radiation in Modeling of Induction Heating Systems Jerzy Barglik, Michał Czerwiński, Mieczysław Hering and Marcin Wesołowski Time-Domain Analysis of Self-Complementary and Interleaved Log-Periodic Antennas A.X. Lalas, N.V. Kantartzis and T.D. Tsiboukis New Spherical Resonant Actuator Y. Hasegawa, T. Yamamoto, K. Hirata, Y. Mitsutake and T. Ota

202

212 220

Chapter C. Applications C1. Electrical Machines and Transformers Influence of the Correlated Location of Cores of TPZ Class Protective Current Transformers on Their Transient State Parameters Elzbieta Lesniewska and Wieslaw Jalmuzny Machine with a Rotor Structure Supported Only by Buried Magnets Jere Kolehmainen

231 240

FEM Study of the Rotor Slot Design Influences on the Induction Machine Characteristics Joya Kappatou, Kostas Gyftakis and Athanasios Safacas

247

Concentrated Wound Permanent Magnet Motors with Different Pole Pair Numbers Pia Salminen, Hanne Jussila, Markku Niemelä and Juha Pyrhönen

253

Air-Gap Magnetic Field of the Unsaturated Slotted Electric Machines Ioan-Adrian Viorel, Larisa Strete, Vasile Iancu and Cosmina Nicula

259

x

Dynamic Simulation of the Transverse Flux Reluctance Linear Motor for Drive Systems Ioan-Adrian Viorel, Larisa Strete and Do-Hyun Kang

268

Influence of Air Gap Diameter to the Performance of Concentrated Wound Permanent Magnet Motors Pia Salminen, Asko Parviainen, Markku Niemelä and Juha Pyrhönen

276

Squirrel-Cage Induction Motor with Intercalated Rotor Slots of Different Geometries V. Fireţeanu

284

Analysis and Performance of a Hybrid Excitation Single-Phase Synchronous Generator Nobuyuki Naoe, Akiyuki Minamide and Kazuya Takemata

294

Numerical Calculation of Eddy Current Losses in Permanent Magnets of BLDC Machine Damijan Miljavec and Bogomir Zidarič

299

Analysis of High Frequency Power Transformer Windings for Leakage Inductance Calculation Mauricio Valencia Ferreira da Luz and Patrick Dular

307

Influence of the Stator Slot Opening Configuration on the Performance of an Axial-Flux Induction Motor Asko Parviainen and Mikko Valtonen

313

Characteristics of Special Linear Induction Motor for LRV Nobuo Fujii, Kentaro Sakata and Takeshi Mizuma

318

Electromagnetic Computations in the End Zone of Power Turbogenerator M. Roytgarts, Yu. Varlamov and А. Smirnov

324

C2. Actuators and Special Devices The Impact of Magnetic Circuit Saturation on Properties of Specially Designed Induction Motor for Polymerization Reactor Andrzej Popenda and Andrzej Rusek

335

Electromagnetic Design of Variable-Reluctance Transducer for Linear Position Sensing J. Corda and S.M. Jamil

343

The Influence of the Matrix Movement in a High Gradient Magnetic Filter on the Critical Temperature Distribution in the Superconducting Coil Antoni Cieśla and Bartłomiej Garda

350

Macro- and Microscopic Approach to the Problem of Distribution of Magnetic Field in the Working Space of the Separator Antoni Cieśla

356

Electric Field Exposure Near the Poles of a MV Line D. Desideri, A. Maschio and E. Poli

363

xi

Study on High Efficiency Swithced Reluctance Drive for Centrifugal Pumping System Jian Li, Junho Cha and Yunhyun Cho

370

C3. Special Applications Power Quality Effects on Ferroresonance Luca Barbieri, Sonia Leva, Vincenzo Maugeri and Adriano P. Morando FEM Computation of Flashover Condition for a Sphere Spark Gap and for a Special Three-Electrode Spark Gap Design Matjaž Gaber and Mladen Trlep Recent Developments in Magnetic Sensing Barbaros Yaman, Sadık Sehit and Ozge Sahin Modelling of Open Magnetic Shields’ Operation to Limit Magnetic Field of High-Current Lines R. Goleman, A. Wac-Włodarczyk, T. Giżewski and D. Czerwiński Selected Problems of the Flux Pinning in HTc Superconductors J. Sosnowski

381

388 396

403 410

The Effect of the Direction of Incident Light on the Frequency Response of p-i-n Photodiodes Jorge Manuel Torres Pereira

417

3-D Finite Element Mesh Optimization Based on a Bacterial Chemotaxis Algorithm S. Coco, A. Laudani, F. Riganti Fulginei and A. Salvini

425

Electro-Quasistatic High Voltage Field Simulation of Large Scale 3D Insulator Structures Including 2D Models for Conductive Pollution Layers Daniel Weida, Thorsten Steinmetz, Markus Clemens, Jens Seifert and Volker Hinrichsen Electromagnetic Aspects of Data Transmission Liliana Byczkowska-Lipińska and Sławomir Wiak

431

438

Application of the Magnetic Field Distribution in Diagnostic Method of Special Construction Wheel Traction Motors Zygmunt Szymański

449

Author Index

457

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Chapter A. Fundamental Problems and Methods A1. Fundamental Problems


Advanced Computer Techniques in Applied Electromagnetics S. Wiak et al. (Eds.) IOS Press, 2008 © 2008 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-58603-895-3-3

3

Power Effect in Magnetic Lamination Taking into Account Elliptical Hysteresis Approach Kazimierz ZAKRZEWSKI Institute of Mechatronics and Information Systems, Technical University of Lodz Stefanowskiego 18/22 str., 90-924 Lodz, Poland [email protected] Abstract. On the grounds the author’s early works the analytical formulae for unit active power (losses) and reactive power in magnetic lamination were presented. The elliptical hysteresis approach of magnetic loops was assumed. In particular, these powers have been refereed to magnetization frequency. The universal functions F1, F2 for the case of magnetic flux forcing and F3, F4 for magnetic strength application on the booth side of lamination have been deduced in the work.

Introduction Despite many efforts, the full satisfactory analytical formulae for active and reactive power in magnetic laminations used in electromagnetic devices and electromechanical converters are not elaborated. The microscopic phenomena which we transferred for macroscopic effects are not exact described analytically. The numerous approximations which model static or dynamic hysteresis loop [1–3] not enable to calculate the power losses or reactive power with exactness acceptable in electrical engineering. Author, for dozen years prefers the use of equivalent elliptical hysteresis loops by approximation of real magnetization characteristics which enable to take account the hysteresis and first kind of eddy current losses. Thanks to introduction so called anomaly coefficient A n the analytical results may be adapt to total losses obtained experimentally [5,6].

Forced Magnetic Flux in Lamination The introduction of elliptic hysteresis loops is connected with assumption that all function of electromagnetic field are sinusoidal in time. The authors formulae deduced in [4] will need for the next discussion. In the paper [4] the active and reactive power were refereed to lamination segment with side surface area equal to 1 m2, in this work will be recalculated for 1 kG of the lamination weight. The measured active power may be expressed Pφ meas = An ( Bav, f )

k3 φm2 ξφ 2 σ μ m2 γ d

(1)

4

K. Zakrzewski / Power Effect in Magnetic Lamination

Table 1. Average values of

An

for ET3 lamination

Bav

T

0,2

0,5

1,0

1,1

1,3

1,5

1,7

An

–

2,0

2,0

1,92

1,76

1,45

1,29

1,29

where: Pφ meas – measured active power, An – anomaly coefficient as function of amplitude Bav (average induction in a cross section of lamination) and frequency f k = π f μ mσ

(2)

μ m – magnetic permeability for Bav, σ – conductivity of lamination material, γ – spe-

cific weight of lamination material, d – thicknees of lamination, φm – forced magnetic flux in lamination cross-section 1 ⋅ d m 2 φm = Bav ⋅ d [Wb/m] ξφ =

a sinh ( akd ) − b sin ( bkd ) cosh ( akd ) − cos ( bkd )

(3)

(4)

a = cos

δ δ + sin 2 2

(5)

b = cos

δ δ − sin 2 2

(6)

δ – angle of elliptical and symmetric hysteresis loop with amplitude Bav. The value An was called anomaly coefficient because takes into account the additional losses in relation to calculated ones, as a result of elliptical and continuous alternating magnetization inside the lamination material. In the work [6], the coefficient An for the transformer lamination ET3 0,35 mm was investigated and described. With a same approximation it is possible to assume the average values of An coefficients in dependence on frequency in a range (5 ÷ 300) Hz as a constant values. The function An ( Bav) is presented in Table 1. The forced magnetic flux appears in laminated core of transformer by voltage excitation. Essential dependence in the transformer praxis is the total losses reference to the frequency as a measurement method for hysteresis losses extrapolation in the case f →0. It will be indicated that frequently used extrapolation by application a direct line is not correct. Using (1)

5


Pφ meas

= An ( Bav )

f

Pφ

(7)

f

where Pφ f

=

π φm2 ( kd ) ξφ . 2μ m d 2γ

(8)

From (8) may be introduce the universal function (not dimensional) F1 = ( kd ) ξφ .

(9)

This formula may be presented in dependence not dimensional value ( kd ) . The 2

relation between f and ( kd ) is 2

f =

( kd )

2

π μm σ d 2

(10)

In the case of reactive power it is possible to introduce the function F2 = ( kd )ψ φ

(11)

and Qφ

=

f

π φm2 ( kd )ψ φ 2 μ m d 2γ

(12)

where ψφ =

b sinh ( akd ) + a sin ( bkd ) cosh ( akd ) − cos ( bkd )

(13)

The reactive power was less investigated than active power. From author’s experience results that an adaptive coefficient for measured power is equal to HI/Hm relation, where HI – first harmonic amplitude, Hm – amplitude of magnetic strength during sinusoidal change of induction with amplitude Bav in conditions of the real hysteresis loop magnetization. Qφ meas f

=

H I Qφ Hm f

The functions F1, F2 are illustrated in Fig. 1.

(14)

6


Figure 1. Diagrams of functions F1, F2.

Forced Magnetic Field Strenght on Lateral Surface of Lamination Another means of practical generation of electromagnetic field in lamination is an excitation by electrical current with forced magnetic field strength on lateral surfaces of lamination. The adequate formulae on the base [4] are for active power (losses) PH π μ m 2 ζ H = Hm f γ ( kd )

(15)

where: ζH =

a sinh ( akd ) − b sin ( bkd ) cosh ( akd ) + cos ( bkd )

(16)

for reactive power QH π μ m 2 ψ H = Hm f γ ( kd )

(17)

where: ψH =

b sinh ( akd ) + a sin ( bkd ) cosh ( akd ) + cos ( bkd )

(18)

Unfortunately, the formulae (15), (17) with adaptation to results of measurements was not yet investigated (open problem).


7

Figure 2. Diagrams of functions F3, F4.

Analogically to F1, F2 it is possible to introduce the universal functions F3 = ζH /(kd), F4 = ψH /(kd) which are illustrated in Fig. 2. Conclusion The universal functions F1 and F2 may be used for investigation of losses and reactive power in a wide range of magnetic flux frequency in lamination (voltage excitation). Adequately, the functions F3 and F4 are interesting for this investigation in a case of forced magnetic field strength on lateral surfaces of lamination (current excitation). The analytical formulae may be helpful in the praxis for design of different electromagnetic devices. References [1] D.C. Jiles, D.L. Atherton: “Theory of Ferromagnetic Hysteresis”, Journal of Magnetism and Magnetic Materials 61 (1986), North-Holland, Amsterdam, pp. 48-60. [2] D. Mayergoyz: “Mathematical Models of Hysteresis”, Springer-Verlag, New York 1991. [3] J.K. Sykulski (editor): “Computational Magnetics” Chapman and Hall, 1995 London, Glasgow, Weinheim, New York, Tokyo, Melbourne. [4] K. Zakrzewski: “Berechnung der Wirk und Blindleistung in einem ferromagnetischen Blach unter Berücksichtigung der Komplexen magnetischen Permeabilität”, Wiss. Z.TH Ilmenau (1970), H.5, s. 101-105. [5] K. Zakrzewski: “Method of calculations of unit power losses and unit reactive power in magnetic laminations in a wide range change of induction and frequency”, Proceedings of ISEF’99 – 12th International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering, ISEF’99, Pavia, Italy, September 23-25 1999, s. 208-211. [6] K. Zakrzewski, W. Kubiak, J. Szulakowski: „Wyznaczanie współczynnika anomalii strat w blachach magnetycznych anizotropowych, Prace Naukowe IMNPE Politechniki Wrocławskiej, Studia i Materiały Nr 20, Wrocław 2000, s. 299-305.

8


Study of Electromagnetic Field Properties in the Neighbourhood of the Metallic Corners Stanisław APANASEWICZ and Stanisław PAWŁOWSKI Technical University in Rzeszów, Chair of Electrodynamics and Electrical Machinery Systems, Poland

Study of electromagnetic field in the neighbourhood of the metallic corners is an aim of the present paper. Concave and convex corners are considered.

Introduction It is known that in the case of depth of electromagnetic field penetration in the metal that is small in comparison with radius of curvature of metal surface, calculation of the field distribution can be simplified. Impedance boundary conditions are applied. Aim of that is simplification of Helmholtz equation in the metallic area: term with second derivative in the direction tangential to the boundary is rejected from this equation and the second derivative in the transverse direction is left. Such a simplified equation is solved in the open way and it causes impedance boundary conditions in the air area. Normal derivative of z component of vector potential A is proportional to A ie ∂A α A = . In the neighbourhood of the corners such simplifications are not possible ∂n μ r and in such cases calculation of electromagnetic field distribution is more complicated. Primary aim established by authors of the present paper is to find adequate simplifications if there is metallic area of large curvature, radius of which is small in comparison to the depth of field penetration. It turned out that this problem looks differently for the concave and convex corners. Therefore, our first task is to study the essential differences occurring in these both cases. At first, we analyze the corners creating angles π/2 lub π/3; however we omit the analysis of rounded corners. We consider three types of field excitation: a) electrostatic, b) incidence of straight flat wave, c) excitation of eddy currents in the metal by fields generated by sinusoidal currents flowing in wires in air area. In the mentioned case, solution of adequate Helmholtz and Laplace equations can be presented in the form of Fourier integrals and determination of integrand comes down to solve an equation of Fredholm integral equations of the second kind. In this paper we are restricted to two first variants.

9

S. Apanasewicz and S. Pawłowski / Study of Electromagnetic Field Properties y

y

−Q

b

γ =

Q

a

a

γ =∞

b

−Q

b

Q

a

x

a

γ =∞

a) Concave corner

γ =0

b

x

b) Convex corner Figure 1. Sketch of studied corners.

Study of Electrostatic Field We assume that electrostatic field is generated by charges Q and –Q placed for the simplicity symmetrically in relation to the metallic walls in points (a,a) and (b,b) (Fig. 1a and 1b). Electrostatic field is described by scalar potential ϕ = ϕ ( x, y ) that fulfils Laplace’s equation. This potential can be presented in the form of two components: ϕ = ϕ 0 + ϕi

(1)

First of them presents the field generated by charges Q, –Q and the second presents influence of metallic walls. Term ϕ0 is known and it is presented in the following way: ϕ0 =

∞ g cosτ x + g 2 sin τ y Q ( x − a)2 + ( y − a)2 Q ln = eτ x 1 dτ , 4πε 0 ( x − b) 2 + ( y − b) 2 2πε 0 ∫0 τ

x < 0 (2)

where: g1 = e − sb cos sb − e − sa cos sa g 2 = e − sb sin sb − e − sa sin sa

However component ϕi can be presented in the following way: ∞

1 ϕi = ∫ D(τ ) ⎡⎣ e−τ x sin τ y + e −τ y sin τ x ⎤⎦ dτ , τ 0

x > 0,

y>0

(3)

10

S. Apanasewicz and S. Pawłowski / Study of Electromagnetic Field Properties

In the case of concave corner, D function is the only function that should be calculated. Potential ϕ must be equal zero on the surface of metal: ϕ ( x, 0) = ϕ (0, y ) = 0 what is tantamount to the zeroing of tangential components of electrostatic field E = ( E1 , E2 , 0) = (−ϕ x , −ϕ y , 0) , so the following condition should be fulfilled ∞

⎤ Q ⎡ x−b x−a − = g ( x) ⎢ 2 2 2 2 ⎥ ( x b ) b ( x a ) a − + − + ⎦ 0 ⎣

∫ D(τ ) cosτ xdτ = 2πε 0

(4)

hence D=

∞

2 g ( x) cosτ xdx π ∫0

(5)

D function and ϕi potential can be calculated in the overt form. Ultimately, omitting laborious computational transformations, we can present solution in the form of mirror images: ϕi ( x, y ) = ϕ∗ (−b, −b) + ϕ∗ (−a, −a ) + ϕ∗ (b, −b) + ϕ∗ (−b, b) − ϕ∗ (a, − a ) − ϕ∗ (− a, a) (6) Q ln ⎡ ( x − a )2 + ( y − b)2 ⎤⎦ . 4πε 0 ⎣ Fictitious Q charges are located in points (a,–a), (b,–b), (–a,–a), (–b,–b), (–a,a), (–b,b). In the event of convex corner in the area x > 0, y > 0 we accept ϕi in the form of Formula (3) and in the area x < 0, y >0:

where ϕ∗ (a, b) =

∞

1 ϕi = ∫ ⎡⎣ R(τ )eτ x sin τ y + R∗ (τ )e −τ y sin τ x ⎤⎦ dτ τ 0

(7)

We assume that ϕi and ϕix are continuous on the line x = 0, y > 0 in order to calculate functions D, R, R*. Additionally, on the line y = 0, x < 0 the condition ϕi + ϕ0 = 0 must be fulfilled. On the basis of these conditions, we obtain: R = D, ∞

∫R

∗

0

hence:

D+R =

cosτ xdτ =

2τ π

Q 2πε 0

∞

∫ [ D( s) − R ( s ) ] s ∗

0

2

ds +τ 2

⎡ ⎤ x −b x−a − = g ( x) , ⎢ 2 2 2 2 ⎥ ( x b ) b ( x a ) a − + − + ⎣ ⎦

(8) x bM

(1) a) The time-step calculation solves the first three Maxwell equations and the con stitutive laws within the magnet in a 2D x-y-coordinate system with H ( x, y, t ) , J ( x, y, t ) , B ( x, y, t ) , E ( x, y, t ) and H = ( H x , H y , 0) , B = ( Bx , By , 0) , J = (0, 0, J z ) , E = (0, 0, Ez ) as rotH = J rotE = −∂B / ∂t

divB = 0

J = κ M,eq E

B = μM H ,

(2)

118

Lj. Petrovic et al. / Numerical Methods for Calculation of Eddy Current Losses

resulting

in

divJ = 0

b

and

the

losses

per

magnet

h

l T M M PM = M ∫ dt ∫ dx ∫ dy ⋅ T 0 0 0

J z2 ( x, y , t ) / κ M,eq , where T = 2π / ω is the longest time period of the calculated eddy

current density. For the time-step calculation the FE program FLUX2D was used [3]. b) The quasi-static method is neglecting rotH = J . With the vector potential A and B = rotA the eddy-current density in each finite element of the mesh is derived from (2) as J = −κ M,eq ⋅ dA / dt .

(3)

c) The semi-analytical post-processing of a magneto-static finite element field solution can be done in different ways. c) (i) A): For “small” magnets (e.g. segmented magnets) with bM < λ/2 the variation of the flux density along the magnet width is replaced by its average value b 1 M B y ( y, t ) = By ( x, y, t )dx . Then the flux variation Φ ( y, t ) = B y ( y , t ) ⋅ bM lM at each bM ∫0 coordinate plane y = const for 0 ≤ y ≤ hM can be given as a Fourier sum Φ ( y, t ) = ∞

∑Φ ( y) ⋅ sin(kω t ) . The eddy current loss formula for a plane material of thickness bM k =1

k

including the eddy current reaction field (4), (5) is applied [5]. If one takes the result only at the top of the magnet y = hM, where the y-component of the field is maximum, one overestimates the losses by a +60% [1], but by averaging with Simpson’s formula (6) the results agree well with methods a) and b). The losses due to Bx are very small, as the flux density in rotor magnet is oriented mainly radial.

PM ( y ) = hM ⋅ bM3 ⋅ lM ⋅

K m,k =

κ M,eq 24

3 shξ k − sinξ k ⋅ ξ k chξ k − cosξ k

∞

⋅ ∑ (kω ) 2 By , k ( y ) 2 K m,k k =1

By , k ( y ) = Φk ( y ) /(bM ⋅ lM ) (4)

ξ k = bM ⋅ μ M ⋅ κ M,eq ⋅ k ⋅ ω / 2

P M = ( PM ( y = 0) + 4 PM ( y = hM / 2) + PM ( y = hM )) / 6

(5)

(6)

c) (i) B): As alternative to (4) for “narrow” magnets the variation of By ( y, t ) = Φ ( y, t ) /(bM ⋅ lM ) with time is taken directly into account, neglecting the influence of the reaction field of eddy currents: 3

PM ( y ) = lM ⋅ hM ⋅

bM ⋅ (d B y ( y, t ) / dt ) 2 ⋅ κ M,eq 12

(7)

c) (ii): For “broad” magnets with bM > λ/2 (e.g. massive magnet pieces per pole) the averaging of the flux density will lead to wrong results. Hence method b) is


a)

b)

119

c)

Figure 1. a) Cross-section of magnet, b) Cross section of Motor A and c) Motor B.

adopted, but as several magneto-static FEM solvers do not give direct access to the vector potential for post-processing, Az had to be reconstructed by x

Az ( x, y, t ) = ∫ By ( x, y, t ) ⋅ dx → J z ( x, y, t ) = −κ M,eq ⋅ dAz ( x, y, t ) / dt ,

(8)

0

before the losses can be evaluated according to the above noted method. In our case the loss evaluation was again done on three circumferential levels 0, hM/2 and hM. Method (ii) appears to be the most exact one of the methods c), because (i) does not give correct values for bM > λ/2. For methods b) and c) the FE program FEMAG was used [6], and for c) alternatively also FLUX2D.

Investigated Machine Topology Two PMSM, called Motor A and Motor B (Fig. 1 b, c), with tooth-coil three-phase winding for 45 kW, 1000/min, 430 Nm, which can operate at constant voltage 400 V up to 3000/min via field weakening with negative d-current, were investigated [1]. With tooth-coil windings the content of MMF harmonics in PMSM is increased, causing increased magnet losses at load. Both machines are water-jacket cooled, and have been built in our lab, featuring 16 poles, a laminated stator and rotor iron stack of the length lFe = 80 mm and NdFeB rotor magnets. Per pole 7 segmented magnets in circumference and 6 in axial direction are used (dimensions bM × lM = 3.6 x 30 mm) with a magnet height hM = 4.8 mm (Motor A) and 4.7 mm (Motor B). The magnet conductivity is κM = 7·105 S/m, the permeability μM = 1.05μ0. Motor A has open stator slots, q = ¼ slots per pole and phase with profiled copper coils on the wider teeth and buried magnets in the rotor. Motor B has semi-closed stator slots, q = ½ slots per pole and phase with round wire copper coils on each tooth of identical width and surface mounted magnets. A second rotor with surface mounted magnet shells of magnet height 4.7 mm and four rows of axial shell length 45 mm with the same pole coverage 0.78 was built, fitting to the stator of Motor B, called Motor B′. For Motor A a rotor with one massive magnet per pole (dimensions bM × lM = 25.2 × 45 mm, four axial magnet rows) was designed (Motor A′), but was only simulated, not manufactured. In FLUX2D the massive magnets per pole are modelled by redefining the face regions of the 7 magnet segments per pole to be 1 entity (Fig. 2 a, b). For Motors A and B κM,eq = 6.25·105 S/m was used, for Motors A′ and B′ κM,eq = 4.37·105 S/m.

120


a)

b)

Figure 2. Calculated eddy current density with Method a) in the magnets at full load 45 kW, 1000/min: a) for Motor B (segmented magnets), b) for Motor B′. Here the 7 magnets are regarded as one magnet entity by the FEM program. Eddy current density range: a) J = –1.3 … + 0.8 A/mm2; b) J = – 1.8 … + 1.1 A/mm2.

Table 1. Calculated losses in the most left magnet segment of a pole for Motor A at 1000 rpm, rated sinusoidal current, 45 kW. Results of Method c) (i) A) with field variation at the top of the magnet (T) and averaged for top, middle and bottom (TMB) of the magnet in comparison to Method a) Method

c) (i) A) T

c) (i) A) TMB

a)

Losses PM,left [W]

0.190

0.097

0.121

Eddy Current Loss Calculation Results in the Permanent Magnets Investigation of Radial Variation of Flux Density Over a Magnet Segment Methods a) and c) (i) A) are compared for Motor A in order to clarify the influence of the change of flux density in radial direction within the “narrow” magnets on the eddy current losses, using the FEM program FLUX2D. The Fourier sum of Method c) (i) A) considered 47 flux harmonics, but already 5 harmonics would have been sufficient. The sinusoidal currents for operation at load are impressed into the stator winding via three current sources. The initial phase angle of the current was adjusted with respect to the initial rotor position to get the rated torque 430 Nm. In Table 1 the calculated losses PM,left are given for the most left magnet segment per pole, for which either the flux variation at the top (label: T) y = hM, or the average of the loss calculation for three levels y = hM (top), y = hM/2 (middle), y = hM (bottom) (label: TMB) were considered. Considering the field variation of By only at the top of the magnet gives by +60% too large losses, as the decrease of the field to the bottom of the magnet is neglected. Therefore the presented loss values in [1] are by about 60% too big. Considering “TMB”, the losses are by 20% too small, as the influence of the field component Bx is neglected. This influence is seen in the central magnet in Fig. 2a. The eddy current losses in all magnets PM of Motor A are shown in Table 2 for no-load (Is = 0) at 1000/min and 3000/min, which vary with n2, hence with a factor 9 in difference. At rated power 45 kW at 1000/min (Isq-current operation) and at 45 kW, 3000/min (field weakening, Isd-Isq-current operation) the losses differ only by a factor 2.4 due to the weakened field at 3000/min. In the same way the losses were evaluated with Method c) (i) B), showing very good coincidence with Method c) (i) A), which proves also numerically, that the influence of the self field of the eddy currents on the losses in segmented magnets with

121


Table 2. As Table 1, but calculated losses for all magnet segments of Motor A Method n [1/min] PM [W]

no-load load

c) (i) A) TMB

c) (i) A) T 1000 0.42

3000 3.8

36

89

a)

1000

3000

1000

3000

0.24 21

2.2 51

0.19 24

1.7 57

Figure 3. Motor B at no load, 1000 rpm: Calculated radial component of flux density in each of seven magnet segments: a) 10 values per magnet segment (dark line), b) 1 average value per magnet segment (light line).

bM < λ/2 is small. Each magnet segment of Motor B was subdivided into 10 finite elements in x-direction to evaluate the flux density profile By(x, t) for a given rotor position (Fig. 3). For example, at no-load Motor B has the dominating wave length of the field harmonic as the slot pitch λ = τQ, which is obviously bigger than bM by a factor of at least 2. The average value By ( y, t ) = Φ ( y, t ) /(bM ⋅ lM ) , which is used for the calculation, is also shown in Fig. 3. Eddy Current Losses in “Broad” Magnets For “broad” magnets bM > λ/2 like in Motor A′ and Motor B′ the calculation Methods c) (i) (either A) or B)) will not give correct results, but yield 40 … 50 (!) times bigger results. This is due to the different eddy current distribution in massive magnets, which is clearly visible in Fig. 2b, where the seven magnet segments are regarded as one piece bM hM

(bM = 25.2 mm, hM = 4.7 mm) with the condition

∫ ∫J 0

z

( x, y ) ⋅ dx ⋅ dy = 0 . Hence only

0

Method a) and b) can be used for the calculation. Both Methods a) and b) consider the influence of the Bx- and By-component on the eddy current losses, but method b) neglects the self field of the eddy currents. Hence it gives about 30% too big results for “broad” magnets (Motor B′) at no-load (Table 3). Alternatively to Method b) the simplified Method c) (ii) may be used, which neglects the influence of Bx and is only averaging the variation of A in the planes y = hM (top), y = hM/2 (middle), y = hM (bottom). At load both Methods a) and b) give nearly the same results. As Method b) was implemented in the FEM program FEMAG [6] and Method a) in the program FLUX2D, there is also a slight influence of the program package itself on this difference. For “narrow” magnets the results of the above described Method c) (i) B) are also close to

122


Table 3. Calculated eddy current magnet losses for Motor B and B′ at no-load (zero current) and at sinusoidal current operation 45 kW; with Method a), Method b) and Methods c) (i) B) and c) (ii) PM [W] Method Motor B no-load Motor B′ no-load Motor B load 45 kW Motor B′ load 45kW

1000 /min c) (ii) c) (i) B) 5.8 7.2 34 – 19 22 223 –

b) 7.6 35 28 207

a) 8.4 27 28 211

b) 68 319 83 454

3000 /min c) (ii) c) (i) B) 52 65 307 – 59 70 489 –

a) 76 240 89 486

Table 4. Inverter-caused losses in the Motors A and A′ for nominal power 45 kW at 1000 /min; Method a) Stator winding supply PM / W

Motor A/Motor A′

Sinusoidal

Inverter (Case 1)

Inverter (Case 2)

25.5/784

33.0/984

32.5/–

Table 5. Fourier analysis of the phase current for inverter-operation of Motor A at 45 kW, 1000 /min fk/Hz

Case

133

665

931

1463

1729

2261

2527

3857

4123

Iˆk /A

1/2

166/157

5.4/12.9

2.2/8.1

1.4/5.1

6.2/7.3

5.1/3.6

1.4/3.8

0.8/2.0

0.9/2.0

the results of Method a). Method c) (ii) gives similar results as Method b), but with a deviation of up to 30%. Due to the reconstruction of the vector potential A from the flux density B it is more inaccurate. Nevertheless this method is helpful, if the used FEM program does not feature any eddy current loss calculation for magnets. Inverter-Caused Eddy Current Losses in the Magnets In the case of inverter supply, the feeding by current sources can be applied only, if the current ripple due to the switching of the inverter is known in advance. If only the inverter data are given, a coupled circuit FE simulation must be used, which is done with Method a). Hence the IGBT transistors of the inverter are modelled in FLUX2D as low resistances in the “on-state” and with high resistance in the “off-state”. Two extremely different sets for these resistance values both for the IGBT transistors (T) and the freewheeling diodes (D) were applied to investigate the influence of transistor data on the eddy current losses in Motor A and Motor A′. The switching frequency was 2 kHz for both cases. Hence the dominant current harmonics occur at frequency side bands around 2 kHz and 4 kHz, but also over-modulation influence is visible. The harmonic spectrum of calculated currents for both cases is given in Table 5 for Motor A. For Motor A′ it was nearly identical. The smaller 1st harmonic in Case 2 influences the result more than the bigger higher harmonics. An increase of eddy current losses due to inverter-caused harmonics of 25 … 30% occurs in both cases. The massive magnets suffer from 30 times (!) higher eddy current losses. Case 1: RON,T = RON,D = 0.1 mΩ, ROFF,T = ROFF,D = 1 MΩ Case 2: RON,T = 50 mΩ, RON,D = 10 mΩ, ROFF,T = 0.1 MΩ, ROFF,D = 50 kΩ. Conclusions The time-stepping Method a) is time-consuming, but allows with a coupled circuit calculation a reliable determination of the eddy current losses in permanent magnets also


123

at inverter supply. The quasi-static position-stepping (Method b)) neglects the influence of the self field of the eddy currents in the magnets. Due to the low conductivity and permeability the penetration depth at the interesting frequencies is bigger than the magnet dimensions. So it delivers nearly the same loss results. In addition three semianalytic post-processing methods to determine eddy current losses are presented (Methods c) (i) A), (i) B), and (ii)). The first two are only valid for “narrow” magnets, where the variation of flux density over the magnet width is small, whereas the third is also useful for “broad” magnets, as long as the self field of the eddy currents is negligible. The influence of the chosen calculation method on the resulting losses is shown for four permanent magnet synchronous machines with a rating 45 kW, 1000/min and tooth coils.

Acknowledgements The authors acknowledge the support of German Research Foundation (DFG) for financing the project FOR575.

References [1] Deak, C; Binder; A.; Magyari, K.: “Magnet Loss Analysis of Permanent-Magnet Synchronous Motors with Concentrated Windings”, ICEM 2006, Chania, Greece, 2006, 6 pages, CD-ROM. [2] Atkinson, G.; Mecrow, B.; Jack, A.; et al.: “The Analysis of Losses in High-Power Fault-Tolerant Machines for Aerospace Applications”, IEEE Trans. Ind. Appl. 42, 2006, p. 1162-1170. [3] FLUX 9.30 User’s Guide, April 2006, www.cedrat.com. [4] Russell, R. L.; Norsworthy, K. H.: “Eddy currents and wall losses in screened-rotor induction motors”, Proc. IEE, p. 163-175, April 1958. [5] Schuisky, W.: Die Berechnung elektrischer Maschinen, Spinger, Wien, 1960. [6] Reichert, K.: FEMAG, interactive program to calculate and analyze 2-dimensional and axis-symmetric Magnetic and Eddy-Current fields, User’s Manual, February 2007, http://people.ee.ethz.ch/~femag.

124


3-D Finite Element Analysis of Interior Permanent Magnet Motors with Stepwise Skewed Rotor Yoshihiro KAWASE a, Tadashi YAMAGUCHI a, Hidetomo SHIOTA a, Kazuo IDA b and Akio YAMAGIWA c a Department of Information Science, Gifu University, 1-1, Yanagido, Gifu, 501-1193, Japan E-mail: [email protected] b Daikin Industries, Ltd., 1000-2, Ohtani, Okamoto-cho, Kusatsu, Shiga, 525-8526, Japan E-mail: [email protected] c Daikin Air-Conditioning And Environmental Laboratory, LTD., 1000-2, Ohtani, Okamoto-cho, Kusatsu, Shiga, 525-8526, Japan E-mail: [email protected] Abstract. In this paper, the effects of the stepwise skew on the torque waveform of an interior permanent magnet motor are analyzed by using the 3-D finite element method. The usefulness of the stepwise skew is confirmed through the calculated torque waveforms and measured ones.

Introduction The interior permanent magnet motors (IPM motors) are widely used as high-efficiency motors in various usage. It is important for the IPM motors to reduce the noise and vibration as well as to improve efficiency [1,2]. In the IPM motors, it is thought that the torque ripple is one of the reasons for the noise and vibration. There are some techniques like the skew of rotor in order to reduce the torque ripple. The interlaminar gap should be considered in the 3-D finite element analysis for skewed IPM motors. Because the axial component of flux density vectors in the cores is computed very large when the interlaminar gap in the cores is not taken into account. In this paper, we analyzed the effects of the stepwise skew of IPM motors on the torque waveform by using the 3-D finite element method (3-D FEM) with gap elements to take the interlaminar gap in the rotor and stator cores into account. The usefulness of the computation is confirmed through the comparison between the calculated torque waveforms and measured ones [3].

Y. Kawase et al. / 3-D Finite Element Analysis of Interior Permanent Magnet Motors

125

Analysis Method Magnetic Field Analysis The fundamental equation of the magnetic field can be written using the magnetic vector potential A as follows: rot( ν rot A ) = J 0

+ ν 0 rot M

(1)

where ν is the reluctivity, J0 is the exciting current density, ν0 is the reluctivity of the vacuum, M is the magnetization of permanent magnet. Gap Elements It is necessary to take into account the interlaminar gap between the electrical steel sheets, which is very small, in order to calculate axial component of the flux density caused by the stepwise skew accurately. However, if the very small air gap is divided by the conventional meshes, it costs a lot of physical memory and CPU time. Therefore, we take into account the interlaminar gap using the gap elements [4]. The weighted residual Gi of Galerkin’s method for the gap elements are given by: G i = D ∫∫

Ss

rot N i (ν 0 rot A ) dS

(2)

where D is the length of the gap, Ss is the region of the gap element, and Ni is the interpolation function. Nodal Force Calculation Nodal force method is to calculate a local magnetic force in the 3-D FEM. The force Fn on each node n can be calculated as follows [5]: Fn = −

∫

V

(3)

(T grad N n ) d V

where V is the total volume of elements related the node n, T is the Maxwell stress tensor, and Nn is the interpolation function of elements related the node n. The torque Tm is given as follows: Tm =

R×r ∑ ( Fn ⋅ λ r ) , λ = Ω

R×r

(4)

where Ω is all the nodes contained in the rotor region (the rotor and the half of air gap), λ is the unit vector of a rotary direction, r is the directional vector towards the node n, and R is the axis directional vector [6].

126


(a) rotor core and magnets

(b) stator core and coil

Figure 1. Photographs of an IPM motor.

(a) no-skewed rotor model

Figure 2. Analyzed model (no-skewed rotor model).

(c) skewed rotor model B (4 magnets)

(b) skewed rotor model A (2 magnets)

Figure 3. Appearances of stepwise skew.

Figure 4. 3-D finite element mesh (except coil and air).

Table 1. Analysis conditions Number of poles Exciting current (A) Coil Number of turns (turn/slot) Magnetization of magnet (T) Frequency of coil current (Hz) Revolution speed (min–1)

4 14 22 1.2 60 1,800

Analyzed Model and Conditions Figure 1 shows the photographs of an IPM motor. Figure 2 shows the analyzed model of an IPM motor. Figure 3 shows the appearances of stepwise skew. Figure 3(a) shows the no-skewed rotor model. Figures 3(b) and 3(c) show the stepwise skewed rotor model. Figure 4 shows the 3-D finite element mesh. The gap elements are inserted in each x-y plane in the stator and rotor cores. It is assumed that the space factor of the electrical steel sheets of the stator and rotor core is 96%. The analyzed region is 1/4 of the whole region because of the symmetry and the periodicity. Table 1 shows the analysis conditions.


127

Figure 5. Distributions of flux density vectors (o-a section).

Results and Discussions Figure 5 shows the distributions of flux density vectors in o-a section. The distributions of flux density vectors of Figs 5(i) and 5(ii) in each model look almost the same by comparing between no-skewed rotor model and with skewed one. Therefore, Fig. 5(iii) shows the difference between Figs. 5(i) and 5(ii). From Fig. 5(iii), it is found that the axial components of flux density vectors between stepwise skewed magnets of rotor core are very large. Figure 6 shows the waveforms of cogging torque with and without gap elements. In the no-skewed rotor model, the waveforms of cogging torque with gap elements and without one is about the same. On the other hand, in the skewed rotor model, it is found that the peak of cogging torque is different between with gap elements and without one. Consequently, it is found that the usufulness of considering gap elements is clarified.

128


Figure 6. Waveforms of cogging torque with and without gap elements.

Figure 7. Waveforms of torque. Table 2. Discretization data and CPU time Number of elements

814,716

Number of nodes

141,470

Number of edges

967,355

Number of unknown variables

939,928

Number of time steps

180

Computer used: Pentium 4 (3.0GHz) PC

Figure 7 shows the waveforms of torque, which is normalized by the average of the torque of no-skewed rotor model. From Fig. 7, it is found that the calculated waveforms of torque agree very well with measured ones. It is also found that the torque ripples of skewed rotor models A and B are reduced to about 1.5% of the no-skewed rotor model by the stepwise skew in both calculation and measurement. Table 2 shows the discretization data and CPU time.


129

Conclusion The effects of the stepwise skew of IPM motors on the torque waveforms were analyzed using the 3-D FEM with gap elements to take the interlaminar gap in the rotor and stator cores into account. It was found that the ripples of torque can be reduced by the stepwise skew. It was also found that there is ample effect for the stepwise skew by 2 magnets to reduce the torque ripple in this model. The validity of the stepwise skew was confirmed through the calculations and measurements.

References [1] Y. Kawase, T. Yamaguchi, S. Sano, M. Igata, K. Ida and A. Yamagiwa, “3-D eddy current analysis in a silicon steel sheet of an interior permanent magnet motor”, IEEE Trans. on Magnetics, vol. 39, no. 3, pp. 1448-1451, May, 2003. [2] Y. Kawase, N. Mimura and K. Ida, “3-D electromagnetic force analysis of effects of off-center of rotor in interior permanent magnet synchronous motor”, IEEE Trans. on Magnetics, vol. 36, no. 4, pp. 1858-1862, July, 2000. [3] A. Yamagiwa, K. Nishijima, Y. Sanga, Y. Kawase, T. Yamaguchi and T. Yano, “Reduction of Motor Vibration by Stepwise Skewed Rotor”, Japan Industry Applications Society Conference, No. 3-92, 2005. [4] T. Nakata, N. Takahashi, K. Fujiwara and Y. Shiraki, “3-D magnetic field analysis using special elements”, IEEE Trans. Magn. vol. 26, no. 5, pp. 2379-2381, 1990. [5] A. Kameari, “Local force calculation in 3D FEM with edge elements”, International Journal of Applied Electromagnetics in Materials, vol. 3, pp. 231-240, 1993. [6] Y. Kawase, H. Kikuchi and S. Ito, “3-D Nonlinear Transient Analysis of Dynamic Behavior of the Clapper Type DC Electromagnet”, IEEE Trans. Magn. vol. 27, no. 5, pp. 4238-4241, 1991.

130


Advance Computer Techniques in Modelling of High-Speed Induction Motor Maria DEMS and Krzysztof KOMĘZA Institute of Mechatronics and Information Systems, Technical University of Lodz, ul. Stefanowskiego 18/22, 90-924 Lodz, Poland E-mail: [email protected], [email protected] Abstract. In the paper, circuit and field-circuit analyses of high-speed small size induction motors are presented. The circuit analysis is possible only for the first harmonic of the supply voltage. For the real shape of the voltage which has many higher harmonics accurate field – circuit analysis is necessary, but this method is very time consuming. The circuit and field-circuit analyses were done for 2-D structure of the motor and for all values of applied frequencies. The results of calculation of magnetizing current are compared with the measurement.

Introduction Nowadays the high-speed induction motors are widely used in many industrial installations and also in aircraft industry. Many of them are designed as converter-fed induction machines. Some electrical drives with not so sophisticated speed control have voltage shape which has many higher harmonics. The field-circuit method makes accurate computation of high-speed induction motors characteristics possible. Unfortunately, this method is very time consuming. When the motor is supplied by a PWM inverter the length of time step must be smaller than time given by carrier frequency of the inverter. Furthermore the Newton-Raphson iteration is carried out at each time step to consider the magnetic saturation. Therefore in design and optimisation process of these motors improved classical circuit methods are very interesting. In the paper, the different field-circuit and circuit methods are presented and the results are compared with the measurement.

Object of Investigation A high-speed construction of the small induction motor was designed basing on the classical structure of the four-pole induction motor model size 80. The supply voltage was 230 V for the frequency 200 Hz and stator windings were delta connected. The number of series turns of stator windings was 216. This motor had stator core shape with cut of parts making stator yoke width not constant; the external maximal diameter of the stator core was Dse max = 120 mm, and external minimal diameter Dse min = 114 mm [ ]. In the field – circuit analysis the real shape of the stator core was taken into account, but in the circuit analysis the calculations were made for the average value of the external stator diameter Dse av = 117 mm, The motor construction is with closed rotor slots.

M. Dems and K. Kom˛eza / Modelling of High-Speed Induction Motor

131

Field-Circuit Analysis of the Motor The field-circuit analysis of the high-speed induction motors can be made using different levels of accuracy. For each circuit, including the parts described with the field equations and external elements described with the resistance Rz and inductance Lz, the circuit equation has the following form: u = R zi + Lz

di + ∑ d j ΔV j dt j

(1)

Applying this to the combination of the presented equations leads to the following system: ⎡G H ⎢ ⎢0 W ⎢⎣ 0 D T

0 ⎤⎡ A ⎤ ⎡ Q ⎥⎢ ⎥ ⎢ D ⎥ ⎢ΔV ⎥ + ⎢H T R ⎥⎦ ⎢⎣ i ⎥⎦ ⎢⎣ 0

0

0⎤ ∂ 0 0 ⎥⎥ ∂t 0 L ⎥⎦

⎡ A ⎤ ⎡J⎤ ⎢ΔV ⎥ = ⎢ 0 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ i ⎥⎦ ⎢⎣U ⎥⎦

(2)

where

Gij = ∫

1 ∇N i ∇N j dS μ

H ij = ∫ γ N i dS k

Wkk = ∫ γ dS k

Qij = ∫ γ N i N j dS

J i = ∫ J 0 N i dS (3) R and L are diagonal matrixes of the resistances and external inductances. The equations system can be presented in a more general form: RX + S

∂X =B ∂t

(4)

For quasi-static model the assumption that all field variables are varying sinusoidally is made. Since the potential and the currents are varying sinusoidally, they can be expressed as the real part of complex functions. The equations system now becomes: RX + iSX = B

(5)

This system can be solved using complex arithmetic. The quasi-static solvers calculate element permeability using amplitude of the magnetic flux density. This can introduce some errors in highly saturated small machines despite the transient calculation of the magnetisation current needed. In this case, the system can be solved using differential schema with time step equal to Θ

S⎤ S⎤ ⎡ ⎡ ⎢ R (1 − Θ ) − Δt ⎥ X n + ⎢ RΘ + Δt ⎥ X n +1 + (1 − Θ )B n + ΘB n +1 = 0 ⎦ ⎦ ⎣ ⎣

(6)

132


Figure 1. Vector plot and distribution of magnitude of magnetic flux density for 200 Hz supply. current [A] 3 calculated 2 measured 1

0 0,1632 -1

0,1642

0,1652

0,1662

0,1672

0,1682

0,1692 time [s]

-2

-3

Figure 2. Comparison of calculated and measured current values versus time for 200 Hz supply.

Nowadays, the economy of the production causes the use of the construction with not round stator core in motor manufacturing. This type of construction has very high saturation of the motor stator yoke parts with decreased width. Additional problems appeared when the motor has been supplied by the PWM inverter. In this case the supply voltages have complicated shape which depends on the type and algorithm used by the control system. In this case two methods are available: the first when only the first harmonic of the supply voltage is taken into account and the second when the real voltage shape is used. The last method is of course the most accurate but is very time consuming according to the fact that the transient state has to be considered before the steady state is achieved. Figure 1 shows the vector plot and magnitude of magnetic flux density calculated by mentioned methods for 200 Hz supply. It can be noticed that the magnitude of magnetic flux density for this case is rather small despite the fact that above 120 Hz the voltage is constant. Figure 2 presents the calculated terminal current values versus time for real voltage shape at the motor compared with measured values, for 200 Hz supply. The main


133

Figure 3. Vector plot and distribution of magnitude of magnetic flux density for 100 Hz supply, for the first time instant.

Figure 4. Vector plot and distribution of magnitude of magnetic flux density for 100 Hz supply, for the second time instant.

source of differences is the influence of capacitances they are not incorporated in fieldcircuit model. In Figs 3 and 4, two different times instant for 100 Hz supply are presented. From these it can be seen that saturation of the motor changes significantly in time depend on the resultant flux position. In Fig. 5 the voltage and current versus time, calculated and measured, for 100 Hz supply are presented. In this Figure, the first harmonic of the supply voltage is also presented. Values of no-load currents, calculated with the first harmonic supply, by field-circuit and circuit methods correspond roughly. The comparisons with the measurement show that the current values calculated with this assumption are very narrow.

134 800

M. Dems and K. Kom˛eza / Modelling of High-Speed Induction Motor 5

voltage [V]

current [A]

measured

4

600

3 400

2 200

1

0 0,18

0,19

0,2

0,21

0,22

0,23

-200

time [s]

0 0,066 -1

0,068

0,07

0,072

0,074

0,076

0,078

0,08

-2

-400

calculated

-3 -600 time [s] -800

-4 -5

Figure 5. Voltage and first harmonic of the voltage and comparison of calculated and measured current values versus time for 100 Hz supply. Table 1. Flux density in the motor core for different values of the frequency 50 Hz

100Hz

150 H

200Hz

Flux density in the stator yoke [T]

Frequency

1,674

1,709

1,376

1,031

Flux density in the stator tooth [T]

1,481

1,476

1,185

0,892

Flux density in the rotor tooth [T]

1,478

1,473

1,182

0,890

Flux density in the rotor yoke [T]

0,828

0,825

0,662

0,497

effective voltage [V] 300

magnetizing current [A] 2,5

250

2,0

200

measured

1,5

first harmonic of the voltage

150

1,0

measured

calculated for first harmonic of the voltage

100 0,5

50

frequency [Hz]

frequency [Hz] 0,0

0 0

20

40

60

80

100

120

140

160

180

200

0

20

40

60

80

100

120

140

160

180

200

Figure 6. Effective voltage and magnetizing current vs. frequency measured and calculated for first harmonic of the supply voltage.

Calculation of Magnetizing Current of the Motor Using Circuit Method For this motor the calculation using improved circuit method enables the calculation for higher frequencies of the magnetising current for different conditions of the supply of the motor. From 10 Hz to 120 Hz the linear increase of the supply voltage and frequency was made, and in result we obtain the value of flux density, and torque equal constant. For the frequency higher than 120 Hz the value of the voltage was constant. The values of magnetic flux density in the motor core calculated using the circuit method (STAT) are shown in Table 1. Figure 6 shows the effective value of supply voltage, measured, and the first harmonic of this voltage, which was used for calculations. In this figure also magnetizing current of the motor, measured and calculated for first harmonic of the supply voltage was shown.

135

M. Dems and K. Kom˛eza / Modelling of High-Speed Induction Motor power factor 0,8

current [A] 6,0

100Hz, 120Hz

0,7

5,0 4,0

0,5

3,0

30Hz

50Hz

0,4

0,2

1,0

0,1 relative power output 0,25

0,50

0,75

1,00

1,25

1,50

0,0 0,00

0,50

0,75

1,00

1,25

1,50

0,10

100Hz, 120 Hz

0,09

0,7

0,08

0,6

0,07

0,5

50Hz

0,4

30Hz

0,06

30H

0,05 0,04

0,3

0,03

0,2

100Hz

120Hz

50Hz

0,02

0,1 0,0 0,00

relative power output 0,25

slip

efficiency 0,9 0,8

100Hz, 120Hz

0,3

2,0

0,0 0,00

50H 30H

0,6


0,50

0,75

1,00

1,25

1,50

0,01 0,00 0,00


0,50

0,75

1,00

1,25

1,50

Figure 7. Current, power factor, efficiency and rotor slip versus output power for the high-speed induction motor, for different values of the frequency, for linear increase supply voltage.

Computing of Operating Curves Using Circuit Model For this induction motor the parameters and curves of current, power factor, efficiency and rotor slip versus output power were computed, for different values of the frequency. The results of these calculations for the different values of the frequency for linear increase of the supply voltage are shown in Fig. 7. Figure 8 shows the same curves for the frequency higher than 120 Hz and constant value of the supply voltage. From Fig. 7 and Fig. 8 we can state that the highest value of the efficiency and power factor we obtain for the frequency equal to 200 Hz. It is caused by the lowest value of the magnetising current and in result also total stator current of the motor.

Conclusion In the calculations of the high-speed small power induction motors supplied by inverter the higher harmonics and also nonlinear phenomena can be taken into account only in the field-circuit models but they are still very time consuming. Therefore, in design and optimisation process of these motors improved classical circuit methods are very interesting, but their use is possible for the first harmonic of the supply voltage only.

136


current [A]

power factor 0,8

6,0

120Hz

0,6

4,0

0,5 0,4

3,0 150H

2,0

0,50

0,1

0,75

1,00

1,25

1,50

100Hz


0,50

0,75

1,00

1,25

1,50

0,030

150Hz

0,025

0,7 0,6

200Hz 150Hz

0,020

0,5

200Hz

0,4

0,015

0,3

0,010

0,2

120Hz

0,005

0,1 0,0 0,00

0,0 0,00 slip

efficiency 0,9 0,8

0,2 relative power output

0,25

120Hz

150Hz

0,3

200Hz

1,0 0,0 0,00

200Hz

0,7

5,0


0,50

0,75

1,00

1,25

1,50

0,000 0,00


0,50

0,75

1,00

1,25

1,50

Figure 8. Current, power factor, efficiency and rotor slip versus output power for the high-speed induction motor, for different values of the frequency, for constant value of the supply voltage.

References [1] Dems M., Komęza K, Wiak S., Stec T., Kikosicki M., Application of circuit and field-circuit methods in designing process of small induction motors with stator cores made from amorphous iron, COMPEL, The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, vol. 25, No. 2, 2006, s. 283-296. [2] Dems M., Modeling of electromechanical transient processes in induction motors with closed slots of the rotor, Archives of Electrical Engineering, nr. 3, 1997, pp. 333-353. [3] Dems M., Komęza K., A comparison of circuit and field-circuit models of electromechanical transient processes of the induction motor with power controller supply, Proceedings COMPUMAG’2001, Lyon– Evian, France, 2-5 July, 2001, pp. 206-207. [4] Dems M., Rutkowski Z., “STATz_F Software for calculation of electromagnetic parameters and characteristics of induction motors”, Technical University of Lodz, Poland. [5] Gąsiorowski T., “Experiences of FSE “BESEL” S.A. in production of induction motors supplied from frequency inverters”, Proceedings of VII Symposium PPEE 99, Ustroń 22-25 mars 1999, pp. 2-6. [6] PC OPERA-2D – version 11, Software for electromagnetic design from Vector Fields, 2006.


137

Computation of the Equivalent Characteristics of Anisotropic Laminated Magnetic Cores E. NAPIERALSKA-JUSZCZAK, D. ROGER, S. DUCHESNE and J.-Ph. LECOINTE Laboratoire Systèmes Electrotechniques et Environnement, Technoparc Futura, 62400 Béthune, France [email protected] Abstract. The method presented in this paper allows computing the equivalent characteristic of anisotropic steel sheets used to stack the magnetic circuits. The reluctivity of the anisotropic sheets is a function of the flux density B and the angle between the flux density vector and the rolling direction. This paper focuses particularly the problem of modeling different kinds of overlaps and apparent air-gaps is solved by the homogenization technique which is based on the assumption that, in the layer structure, the magnetic field energy tends to achieve its minimum. The presented method has been verified by comparing the computational results with the measurements of real sets. The paper presents the analysis of magnetic properties of overlaps taking into account the magnetic characteristics of the steel.

1. Introduction The effects caused by magnetic fluxes in anisotropic cores are difficult to handle, both theoretically and numerically. Modeling of the overlapping regions of thin laminations, of thickness 0.2–0.3 mm and surface area of tens or hundreds of square centimeters, taking account of the insulation of micrometer thickness, is practically impossible for realistic systems and thus it poses significant challenges. The authors have undertaken research related to such anisotropic magnetic cores used in transformers and electrical machines. In particular, they are addressing the issues of the influence of the position and pattern of equivalent air-gaps, under different overlapping arrangements, on iron losses and noise due to magnetostriction. This research programme necessitates creation of several mathematical models capable of simulating the transfer of flux between laminations under different anisotropy angles and various air-gap positions. This paper presents the results from the first stage of the project. A homogenization technique has been developped to approximate the overlapping of the laminations. This has enabled to replace the real three-dimensional structures by far simpler homogenous two-dimensional models. The approach relies on the natural tendency of the energy of the magnetic field to achieve a minimum in the non-homogenous laminated structure [1–4]. When calculating the flux density vector in each lamination using the minimisation principle, the functional has been assumed in terms of the energy in the whole structure, while the constraint is the relationship between the flux density vector in the whole structure and the relevant vectors in every lamination. At last, the distribution (sub-division) of the total flux density vector between component laminations of a particular overlapping structure will lead to:

138

E. Napieralska-Juszczak et al. / Characteristics of Anisotropic Laminated Magnetic Cores

• • • •

a method of calculating the equivalent (homogenised) reluctivity at any point of the laminated sheet and/or the air-gap; a method of establishing equivalent characteristics for various structures; a method of calculating the magnetic field distribution in a transformer core under different overlapping schemes; a method of calculating the magnetic field distribution in an electrical machine, when the core is anisotropic, at various anisotropy angles.

2. Method Presentation Using the homogenization technique makes it possible to replace real 3D structures by simpler homogeneous 2D structures. In the homogenization technique, concepts of macrostructure and microstructure are of importance. In the considered problem, the macrostructure is a complete assembly of layers, while the microstructure is a repeatable structure of two or more layers made of sheets with different rolling directions. In the case of apparent air gaps, the microstructure is a set of sheet layers and air layers. The usual procedure consists in defining a basic volume V able to replace the whole structure [5,6]. Thus, the main purpose in this paper is to replace the set of microstructures by the equivalent volume V without changing the flux distribution in each layer. The first step of the method is to define the components of the resulting flux den sity vector B . The relation between B and the flux density vectors b in the repetitive structures representing a full set of the overlapped layers is given by formula (1) where V is the macrostructure basic volume.

1 B = ∫ bdv VV

(1)

If the integral is replaced by the sum for the basic volume dx*dy*H, Eq. (2) is obtained. Denotes n is the number of different not repetitive layers in the structure, hi is the thickness of the layer i and H is the thickness of the macrostructure. B=

1 dx ⋅ dy ⋅ H

n

∑ B ⋅ dx ⋅ dy ⋅ h i =1

i

i

(2)

The components Bix and Biy (Eq. 3a and 3b) of the flux density B in point P of a multilayer set representing overlapping of sheets (i = 1, 2, 3,…n) are computed. The calculation is based on the total flux penetrating through the limiting surfaces of the volume V. Bix =

Biy =

φix′ + φix′′ 2 ⋅ dy ⋅ hi φiy′ + φiy′′ 2 ⋅ dx ⋅ hi

(3a)

(3b)


139

φ’’y

φ’x

φ’’x φ’y

axis x

Figure 1. Fluxes going inside and outside the volume V.

φix′ and φix′ are the fluxes going inside the volume; φix′′ and φix′′ are the fluxes going outside the volume (Fig. 1). The fluxes going through the limiting surfaces of the volume V have to be equal to the sum of fluxes in all particular layers. Therefore the components Bx and By of the resulting flux density vector are given by Eqs (4a) and (4b). The flux density component in the direction z is smaller then 10% of Bx and By, so this component can be omitted. n

Bx =

∑ φ ′ + φ ′′ i =1

ix

ix

(4a)

2 ⋅ dy ⋅ H n

By =

∑ φ ′ + φ ′′ i =1

iy

iy

(4b)

2 ⋅ dx ⋅ H

In order to replace the three-dimensional system by a 2D system, it is assumed that the analyzed volume is made of an equivalent homogenous material in which the distribution of magnetic field is identical to the resulting distribution in the real structure. The second step of the method is to define the goal function to minimize the magnetic energy. The total energy of the macrostructure results from the sum (Eq. 5) of the magnetic energy Wµi stored in each volume Vlay of the layer i (Eq. 6). µi and Bi are respectively the permeability and the flux density vector of the layer i. Wµ min = ∑ Wµi µi , Bi

(

1 Wμi = Vlay Bi H i 2

)

(5)

(6)

140


To calculate flux density vectors in the particular layer, a minimization method is applied. The applied goal function is the minimum of the magnetic energy in the volume V, while the relationship between the resultant flux density vector and each vector of the flux density in particular layers is the restriction. The goal function depends on the computed structure and it has to be prepared taking into account the interlacing, the insulation between the laminations and the air-gaps. On the one hand, the air-gaps depend on the inaccuracy of the sheet overlapping. On the other hand, it is possible to take advantage of the air gaps to get the required orientation of the magnetic flux. Indeed, apparent air-gaps exist on the edges and inside the structure made of overlapped sheets. Air gaps and sheets form a non-homogenous layer structure. That is why two kinds of materials – steel sheets with permeability μFe and air with permeability μ0 – appear in parallel, forming together the apparent air gap. The division of the resulting flux density B between the steel and air depends as well on the rule of minimum of energy stored in magnetic field. The resulting flux density vector in apparent air-gap is given by the expression (7) were B0 is the flux density in the air, BFe is the flux density in the sheet.

Bres = BFe + B0

(7)

The magnetic energy stored in the structure results from the sum of the energy stored in the air gaps and in the sheets. If Vlay denotes the volume of the simple layer, nFe the number of sheet layers, n0 the number of air layers, ν0 and νFe the reluctivity respectively in air and in sheets, then the goal function for the structure is given by the formula (8). Wµ min BFe , B0

(

)

2 ⎡Vlay nFeν Fe BFe + Vlay n0ν 0 B02 ⎤ =⎢ ⎥ 2 ⎣⎢ ⎦⎥ min

(8)

Third step of the method consists in establishing the family of the anisotropic characteristics of the whole structure. Successive values of the flux density from 0 to the steel saturation are imposed. Every point of the characteristics is determined with the following algorithm. First the flux density vectors for the structure are calculated with the minimization task. Then, the expression (9) makes it possible to calculate the reluctivity of macrostructure. At last, the field intensity H = ν B is calculated. n ν B2 + n ν B2 ν B,Wµ min = Fe Fe Fe 2 0 0 0 nB

(

)

(9)

3. Comparison of Calculation and Measurement Results Calculations for two types of different structures are made. The proposed method is composed the following steps: • •

the homogenized reluctivity at any point of the laminated sheet or the air gap is calculated, the equivalent characteristics are established,


141

Figure 2. Family of characteristics B(H) for the different anisotropy directions.

•

• • •

the properties of the overlapped structure are compared with the magnetic properties of the steel forming these overlaps. It gives, in few seconds, a first idea of the quality of the overlapping without doing a Finite Element (FE) simulation which requires a long computation time, the magnetic field distribution in every tested structure is calculated using a FE method, taking into account a nonlinear anisotropic reluctivity, the division of the total flux density vector between the component laminations is analysed, the results of simulations are compared with measurements.

The first tested structure is made of layers alternatively shifted of 90°: the first layer is placed in the rolling direction (anisotropy angle 0°), the second layer has an anisotropy angle of 90°. This structure corresponds to the conventional two-cycle interlacing (90° transformer core overlap). Figure 2 shows the family of the equivalent magnetizing characteristics B(H). The anisotropy angle α of the whole structure is arbitrary defined as the angle of between the resulting flux density vector and the rolling direction of the first layer. The characteristics of the equivalent structure for α = 0°, 30°, 45°, 60° and 90° are presented at Fig. 2. The characteristic of the anisotropic steel used to form the structure are also presented for the anisotropy angles 0°, 60°, 90°. It makes possible to estimate the quality of the whole structure. Figure 3 presents the total flux density B at the point p(x,y) of the structure (solid line) and the division of B between the layer 1 and the layer 2 (interrupted lines) versus ωt (with ω the grid pulsation). Figure 4 gives the comparison of the calculated results and measurement results for both layers at p(x,y) [8–10]. The calculations were made using measured vectors of the resultant flux density. A second structure makes it possible to test the influence of the air gaps. Its geometry is presented at Fig. 5. Every layer is made of a sheet in the rolling direction (anisotropy angle of 0°), an air gap and a sheet placed perpendicularly to the rolling direction (anisotropy angle of 90°). The position of the air gap and the repartition of the two kinds of magnetic materials are different for every layer. The characteristics calculated for every layers of this structure and compared with the characteristics of the anisotropic steel are shown at Fig. 6. Figure 7 shows the family of the equivalent magnetizing characteristics B(H) for the layer 4. One can observe that the equivalent layer material has anisotropic properties, like the elementary steel

142


Figure 3. Resultant flux density in the structure divided between layers.

Air gap

90°

Figure 4. Comparison of measurements and calculations.

0°

Layer 1 Layer 2 Layer 3 Layer 4

Figure 5. Presentation of the structure.

Figure 6. Family of characteristic B-H for the directions 0° for the 4 layers.

sheets. The characterization is different for the low and high values of the flux density and the numerical board between them is about 1.1 T. The properties of the equivalent layer are worse than the sheet properties for α between 0° and 45°. However, equivalent layer properties are better if α is superior to 45°. To check the proposed method, not only the simple structure has been tested. Different types of transformers and complex systems have been also studied, for example a transformer supplying a converter system during normal working or working under different kinds of faults.


143

Figure 7. Family of characteristics B(H) for the layer 4 for the different anisotropy directions.

4. Conclusion The measurement results confirm the good accuracy of the proposed method. The average fractional error betweens calculation and measurements is between 5–10%, depending of the structure. This method makes it possible to substitute the complex 3-D structures by far simples two-dimension structures. For example, it gives the influence of the joints and air-gaps on the required magnetic flux distribution. The method can be applied to model all kinds of transformer core overlapping. It allows the calculation of the flux density vectors in any layer of the transformer cores with different overlapped limb and yoke sheets. It should provide a considerable help for the designers since it allows them to arrange cores of the same dimensions but of optimum structure. The presented method can be also applied to design rotating electrical machines equipped with a rotor or a stator made of anisotropic materials. References [1] J. Gyselinck, R.V. Sabariego, P. Dular, “A nonlinear time-domain homogenization technique for laminated iron cores in three-dimensional finite-element models”, IEEE Trans. on mag., Vol. 42, Issue 4, April 2006, pp. 763–766. [2] Hiroyuki Kaimori, Akihisa Kameari, Koji Fujiwara, “FEM Computation of Magnetic Field and Iron Loss in Laminated Iron Core Using Homogenization Method”, IEEE Trans. on mag, Vol. 43, Issue 4, April 2007, pp. 1405–1408. [3] A.J. Bergqvist, S.G. Engdahl, “A homogenization procedure of field quantities in laminated electric steel”, IEEE Trans. on mag, Vol. 37, Issue 5, Part 1, Sept. 2001, pp. 3329–3331. [4] L. Krahenbuhl, P. Dular, T. Zeidan, F. Buret, “Homogenization of lamination stacks in linear magnetodynamics”, IEEE Trans. on Mag, Vol. 40, Issue 2, Part 2, March 2004, pp. 912–915. [5] A. De Rochebrune, J. Dedulle, J. Sabonnadiere, “A Technique of homogenization applied to the modeling of transformers”, IEEE Trans. on Mag., vol. 26, No2, 1991, pp. 520–523. [6] J.M. Dedulle, G. Meunier, A. Foggia, J.C. Sabonnadiere, D. Shen, “Magnetic fields in nonlinear anisotropic grain-oriented iron-sheet”, IEEE Trans. Mag., Vol. 26, N°. 2, 1990, pp. 524–527. [7] E. Napieralska-Juszczak, M. Pietruszka, “Semi-analytical method of modelling the magnetising curves for anisotropic sheets”, 4th Int. Workshop on Electric and Magnetic Fields, Marseille France, 1998, pp. 451–456. [8] M. Pietruszka, E. Napieralska-Juszczak, “Lamination of T-joints in the transformer core”, IEEE Trans. on Mag, Vol. 32, Issue 3, Part 1, May 1996, pp. 1180–1183. [9] A.J. Moses, “Rotational magnetization-problems in experimental and theoretical studies of electrical steels and amorphous magnetic materials”, IEEE Trans. on Mag, Vol. 30, N°2, 1994, pp. 902–906. [10] M. Pietruszka, “A method to compute the magnetic field in anisotropic 3-phase transformer cores with arbitrary overlapping structures”, D.Sc Thesis, Poland, 1995 (ISSN 0137-4834).

144


Improving Solution Time in Obtaining 3D Electric Fields Emanated from High Voltage Power Lines Carlos LEMOS ANTUNES a,b, José CECÍLIO b and Hugo VALENTE c Lab. CAD/CAE, Electrical Engineering Dept., University of Coimbra, Pólo II, 3030 – 290 Coimbra, Portugal b APDEE – Assoc. Port. Prom. Desenv. Eng. Electrotécnica, Rua Eládio Alvarez, Ap. 4102, 3030 – 281 Coimbra, Portugal c REN – Rede Eléctrica Nacional, Av. Estados Unidos da América 55, 1749 – 061 Lisboa, Portugal E-mail: [email protected]; [email protected] a

Abstract. In this paper it is presented an algorithm to reduce the computational time in obtaining the electric field distribution in a plane of analysis due to High Voltage Power Lines. It is used a two dimensional interpolation based on a spline function using as known nodal values, the field solution at nodes of a coarser plan grid.

Introduction The LMAT_SIMEL [1] is a software program that calculates the 3D electric field distribution on specified nodes, emanating from general 3D Line(s) configurations. The electric field is calculated using a 3D integral numerical approach and makes use of the image method. The conductors are considered filamentary wires of arbitrary geometric configuration with known imposed voltages: phase-earth or zero if it corresponds to the guard conductor and the catenary is approximated by straight lineal segments. The electric field can be calculated along any path or on any plane. The earth is considered as a perfect conductor at zero voltage reference value and its influence is taken into account using the method of images. The influence of vegetation and terrain elevations is not taken into consideration. The grid discretization of the solution plane is very important to obtain a good or smoother solution for electric field distribution which may lead to a considerable computational time. To reduce this computational time, we have used a two dimensional interpolation function to estimate the field solution at intermediate nodes, from the field solution obtained in a coarser plane grid.

Formulation The phasor electric field Eˆ at any point P(x, y, z) due to a Line, is calculated by

Eˆ =

C. Lemos Antunes et al. / Improving Solution Time in Obtaining 3D Electric Fields

145

ˆ (s) 1 λ 1 NS i ⋅ ∑ Li ⋅ ∫ ⋅ ds ⋅ aˆ 2 0 4πε 0 i =1 r − r'

(1)

where the point P(x, y, z) is defined by r and the phasor charge density λî in the seg-

(

)

ment i is located at r ' , with aˆ as the unit vector in direction r − r ' . It is seen that the phasor linear charge density has to be previously calculated for all the Line(s) and their images. For each line segment the charge distribution is approached by a cubic spline polynomial as (2).

λˆ ( s ) = c0 + c1s + c2 s 2 + c3 s 3

(2)

where ⎛ 3s 2 2 s 3 ⎞ ⎛ ⎛ 3s 2 2 s 3 ⎞ ⎛ s 2 s3 ⎞ 2s 2 s 3 ⎞ c0 = ⎜1 − 2 + 3 ⎟ , c1 = ⎜ s − + 2 ⎟ , c2 = ⎜ 2 − 3 ⎟ and c3 = ⎜ − + 2 ⎟ L1 L1 ⎠ L1 L1 ⎠ L1 ⎠ ⎝ ⎝ ⎝ L1 ⎝ L1 L1 ⎠ s is an adimensional parameter ( s = 0 at the beginning of the segment and s = 1 at the end of that segment). For each Line node i it is required the calculation of the phasor charge density λî and its derivative λˆ' , which is based on the following equation: i

VˆP =

t 4πε 0

∫

⎡ˆ ⎣ λ0 1

0

⎡k ( s)⎤ λˆ1 ⎦⎤ ⋅ ⎢ o ⎥ ⎣ k1 ( S ) ⎦ ds + t 4πε 0 r − r'

∫

1

0

⎡ ˆ' ⎣ λ0

⎡ k0' ( s ) ⎤ λˆ1' ⎦⎤ ⋅ ⎢ ' ⎥ ⎢⎣ k1 ( S ) ⎦⎥ ds r − r'

(3)

It was used a two dimensional interpolation function [2,3] to estimate the intermediate values from two known values. The interpolation is essential to obtain a smoother or good representation of electric field distribution, between two known field values. Given a rectangular grid { xk , yl } and the associated set of numbers zkl which correspond to the known field values, with 1 ≤ k ≤ m, 1 ≤ l ≤ n , we have to find a bivariate function z = f ( x, y ) that interpolates the data (field solution), i.e., f ( xk , yl ) = zkl for all values of k and l . The grid points must be sorted monotonically, i.e. x1 < x2 < ⋅⋅⋅ < xm with a similar ordering of the y-ordinates. To generate a bivariate interpolation on the rectangular grids and calculate the value in the points specified in the arrays xi and yi it is used a spline interpolation, like (4), for example for x: P ( x ) = a ⋅ x3 + b ⋅ x 2 + c ⋅ x + d

(4)

The corresponding mathematical spline must have a continuous second derivative and satisfy the same interpolation constraints. The breakpoints of a spline are also referred to as knots.

146

C. Lemos Antunes et al. / Improving Solution Time in Obtaining 3D Electric Fields Field distribution in the plane

Field distribution In the plane

4000

5000 4000

3000

Eef

Eef

3000 2000

1000

2000 1000

0 100

0 100 100 80

50

60

100 80

50

60

40 y

0

40

20 0

20 x

Figure 1. Electric Field smoother solution (Case 1).

y

0

0

x

Figure 2. Electric Field smoother solution (Case 2).

The first derivative P' ( x ) of our piecewise cubic function is defined by different formulas on either side of a knot xk . Both formulas yield the same value d k at the knots, so P' ( x ) is continuous.

Case Studies It is presented as illustration examples two case studies regarding the electric field emanated from High Voltage Power Lines. Case 1 corresponds to a single Line and Case 2 corresponds to two Lines orthogonally placed to each other. For both cases the electrical conditions are the same, with 220 kV and 1140 A per phase conductor and the catenary of the Line(s) approached by 30 straight lineal segments. Both Lines have 100 m length. The solution plane is defined by a span of Line and it was considered as reference, a grid defined by one meter space between nodes in the solution plane. This discretization corresponds to a grid with 10000 nodes and produces a smoother solution of the electric field. It is seen in Fig. 1 the electric field distribution (smoother solution) in the solution plane for Case 1, and in Fig. 2 for Case 2. To obtain the electric field for Case 1 and Case 2, with 10000 nodes a considerable computational time was required namely 32.49 ×103 sec and 6.51× 104 sec ≈18.2 hours respectively. The computational time to obtain the charge density and the corresponding derivatives has to be added to this time for obtain the total computational time. The idea was then to produce an electric field solution as accurate as possible but with considerable much less computational time.

Results Three different grids with two meter space, five meter space and ten meter space between nodes were used. The field solutions at these nodes were exactly the same as for the finer grid and the derived field solution for the other nodes of the finer grid were processed by the interpolation function. It is shown in Fig. 3 the different computational time t versus the nº of nodes n for Case 1.


147

4

3.5

x 10

3

t=Computational time [s]

2.5

2

1.5

1

0.5

0

0

1000

2000

3000

4000 5000 6000 n=nº of nodes

7000

8000

9000 10000

Figure 3. Computational time (Case 1). Error (%) - 10x10 grid 100

2.52e-001

90 2.10e-001 80 70 1.68e-001

y

60 50

1.26e-001

40 8.41e-002 30 20 4.21e-002 10 0

0

10

20

30

40

50 x

60

70

80

90

100

0

Figure 4. 2D projection of the error distribution in plane (Case 1).

The computational time function t = t ( n ) can be approximated by Eq. 5. t (n) = 3.2425 ⋅ n + 60.53

(5)

To access the accuracy of the field solution, a nodal local error parameter ε n [ % ] was calculated, as: ε n [%] =

En − Eref Eref

(6)

where En is the electric filed value obtained at the nodes for the coarser grid and Eref is the corresponding electric field value obtained with the reference grid (one meter space between nodes). In Fig. 4 it is shown in the form of coloured plot the visualization of a 2D projection of the error distribution at the nodes in the plane of analysis for one grid defined by 10 meter space between nodes for Case 1.

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C. Lemos Antunes et al. / Improving Solution Time in Obtaining 3D Electric Fields Error in x direction for y=70 0.25 2x2 5x5 10x10

0.2

Error [%]

0.15

0.1

0.05

0

0

20

40

60 x

80

100

120

Figure 5. 2D error distribution in plane (Case 1). -3

5

Error in y direction for x=70

x 10

2x2 5x5 10x10

4.5 4 3.5

Error [%]

3 2.5 2 1.5 1 0.5 0

0

20

40

60 x

80

100

120

Figure 6. 2D error distribution in plane (Case 1).

In the form of graphic it is shown in Fig. 5 the variation of ε n [ % ] for a line y = 70 m for the three different grid discretization, and in Fig. 6 it is shown the variation of ε n [ % ] for a line x = 70 m.

As it is seen the error ε n [ % ] varies in the range of [0, 0.252]%, and the computa-

tional time to get the electric field solution is 384.7 sec, thus 84 times lower than the time to obtain the field solution for 10000 nodes. For Case 2, it is shown in Fig. 7 the computational time t versus the nº of nodes n corresponding to these different grids. The computational time function t = t ( n ) can be approximated by Eq. 7. t (n) = 6.4917 ⋅ n + 198.2292

(7)

In Fig. 8 it is shown the visualization of a 2D projection of the error distribution in plane of analysis for one grid defined by 10 meter space between nodes for Case 2.


149

4

7

x 10

6

t=Computational time [s]

5

4

3

2

1

0

0

1000

2000

3000

4000 5000 6000 n=nº of nodes

7000

8000

9000 10000

Figure 7. Computational time (Case 2). Error (%) - 10x10 grid 100

1.00e+001

90 2.79e-001 80 70 2.23e-001

y

60 50

1.67e-001

40 1.12e-001 30 20 5.58e-002 10 0

0

10

20

30

40

50 x

60

70

80

90

100

0

Figure 8. 2D projection of the error distribution in plane (Case 2).

In this case the error ε n [ % ] varies in the range of [0, 10]%. As this error is high, the effective value varies in the range of [0, 500] V/m, it was analysed the error distribution in plane of analysis for one grid defined by 5 meter space between nodes (400 nodes). The 2D projection of this error distribution it is shown in Fig. 9. In this case the error ε n [ % ] varies in the range of [0, 1] %. The nodal error is bigger than for Case 1, but still very low and quite acceptable. The computational time to get the electric field solution is 2.7949 ×103 sec , thus 19 times lower than the time to obtain the field solution for 10000 nodes. The computational time to get the electric field solution for 100 nodes is lower than the time to obtain the field solution for 400 nodes, but the error is bigger. The authors suggest that the user should take as minimal grid configuration, the 5 meter space between nodes (400 nodes) to obtain a quite acceptable field accuracy solution.

150

C. Lemos Antunes et al. / Improving Solution Time in Obtaining 3D Electric Fields Error (%) - 5x5 grid 100

9.96e-001

90 2.77e-002 80 70 2.21e-002

y

60 50

1.66e-002

40 1.11e-002 30 20 5.53e-003 10 0

0

10

20

30

40

50 x

60

70

80

90

100

0

Figure 9. Error distribution in plane for one grid defined by 5 meter space between nodes (Case 2).

Conclusions It was presented an algorithm to reduce the computational time in obtaining the electric field distribution in a plane of analysis due to High Voltage Power Lines. The nodal error in the field solution is quite negligible when comparing solution obtained with finer grids in plane of analysis. This algorithm is implemented in the LMAT_SIMEL software, which is part of a more complete package LMAT_SIMX that allows the analysis and simulation of Electrical and Magnetic Fields emanated from very High Voltage Power Lines, developed by the authors.

Acknowledgments The authors gratefully acknowledge REN-Redes Energéticas Nacionais SGPS, SA for the financial support received under Project COIMBRA_EMF.ELF.

References [1] Carlos Lemos Antunes, José Cecílio, Hugo Valente, “LMAT_SIMEL – The Electric Field numerical calculator of the package LMAT_SIMX for Very High Voltage Power Lines”, accepted for presentation at ISEF 2007 – XIII International Symposium on Electromagnetic Field in Mechatronics, Electrical and Electronic Engineering, Prague, Czech Republic, September 13-15, 2007. [2] www.mathworks.com/access/helpdesk/help/techdoc/ref/index.html. [3] www.mathworks.com.


151

Thermal Distribution Evaluation Directly from the Electromagnetic Field Finite Elements Analysis A. DI NAPOLI, A. LIDOZZI, V. SERRAO and L. SOLERO University ROMA TRE, Via della Vasca Navale 79 – Roma, Italy [email protected] Abstract. This paper deals with the thermal field evaluation achieved directly from the finite elements representation of the relative electromagnetic field. The proposed strategy can be applied to any generic electrical device; in this work it has been proved on a permanent magnets electrical machine. This methodology has been implemented in two different ways. At first laminar and turbulent motions have been considered from thermal convection point of view and then convection has been reduced to pure conduction heat transfer.

Introduction A finite element evaluation algorithm devoted to analyze both electric machines and power switches has been implemented by means of ANSYS@ software. At first the electromagnetic field inside the structure has been evaluated and plotted. The structure has been considered as a discretized surface and both boundary condition and currents have been set. Induction and eddy currents values have been achieved and then dc-current Joule effect losses and the additional losses have been computed. Thermo-electric analysis has been accomplished considering convection and conduction heat transmission. Two different methodologies have been implemented. At first, thermal convection has been studied under the assumption of both laminar and turbulent motion. After that, convection heat transfer has been simulated under laminar motion, where a particular coefficient has allowed reducing the heat exchange to pure conduction, simplifying the simulation model and making the simulation faster. When convection is studied as normal conduction, simulation software ANSYS allows achieving directly the temperature distribution starting from the electric currents, so avoiding previous steps concerning the evaluation of the electromagnetic field. In this manner the simulation time is strongly reduced together with memory occupancy.

Magnetic Analysis Proposed finite elements analysis has been applied to a permanent magnet synchronous machine, where main data are shown in Table 1 and Table 2. Maxwell Eq. (1) written

152

A. di Napoli et al. / Thermal Distribution Evaluation

Table 1. PM machine data Outer diameter Air-gap diameter Air-gap thickness PM residual induction (NdFeB)

D Di

m m m T

δ Bres

0.0125 0.0068 0.001 1.15

Table 2. PM machine evaluated data I

rms

(A) 75 125

Torque (Nm) 20 40

Figure 1. Permanent magnet machine representation and boundary conditions.

Id (A) 69 110

Iq (A) 30 58

Figure 2. Permanent magnet machine mesh.

using the vector A has allowed studying the system as a plane if the observation is far enough from the machine end regions.

μ ∂ A ∇ A= + μJ ρ ∂t 2

(1)

Concerning the boundary conditions, magnetic induction field (B) paths are supposed to be tangent to both inner and outer circumferences shown in Fig. 1 as the segment BD and AC respectively. Finite elements analysis has been implemented to only one by six of the machine section; having the machine three pole-pairs it shows an odd symmetry. Avoiding any saturation effect being the amplitude of the magnetic field quite small, a linear analysis has been performed. Figure 2 shows the achieved mesh useful to evaluate machine inner magnetic fields. Inside the regions where the parasitic currents phenomena are absent, the current density from the previous magnetic fields evaluation has allowed to evaluate the dccurrent losses. In the other regions the heat sources has been accomplished from the eddy currents.


153

Thermal Analysis Thermal analysis is based on the heat sources which can be defined for every mesh point. Machine geometry and its mesh must be kept unchanged when both magnetic and thermal analysis are based on the PLANE77 solver. The data achieved by the magnetic analysis are then used to perform the thermal investigation. Collected data, especially heat sources, have been stored in a data-base and used in the machine mesh. Heat transmission is controlled by the following expression ⋅

q 1 ∂T ∇ T+ = λ α ∂t 2

(2)

where ρ is the mass density [kg/m3], Cp is the specific heat [J/(kgK)], λ is the thermal conductivity [W/(m2K)], q is the sources heat power density [W/m3] and finally α is the thermal diffusion. Along lines AC and BD shown in Fig. 2, boundary conditions have been defined as the expression reported in Eq. (3): −λ

dT = h ( T − T∞ ) dn

(3)

where h takes into account the convection heat transfer from stator outer surface to ambient which is supposed to be T∞ = 30 °C. Surfaces AB and CD, still shown in Fig. 2, have been considered adiabatic. Rotor position affects both power losses and mesh thermal characteristic; temperature is accomplished by the averaging different rotor position since thermal time constant is greater than rotor revolution time. The most difficult task is determining the convection coefficients; these parameters describe heat and mass exchange within the fluid. To this purpose, non-dimensional numbers achieved by analytical and experimental results explain fluid behaviors. Under the assumption of natural convection, convection coefficient can be evaluated starting from the Nusselt number, which can be achieved from Prandlt and Grashof numbers. Convection heat transfer between stator outer surface (BD line in Fig. 3) and airgap has been deeply analyzed where the first surface is usually common with still surfaces. Outer Surface When the heat transfer is based on natural convection, which is true if the stator outer surface and the free air heat exchange is taken into account, main heat transfer numbers are shown as follow. Pr =

Cp ⋅μ λ

= 0.710

(4)

154


Figure 3. Map of magnetic flux, Id = 68.7 A and Iq = 30 A.

Gr =

gβ ( T − T∞ ) D3 ν2

where β =

Figure 4. Map of magnetic flux, Id = 110 A and Iq = 57,9 A.

1 =30 °C T∞

Nu = C ⋅ (Pr⋅ Gr) n

(5)

(6)

where Pr is Prandlt’s number, Gr is the Grashof’s number, Nu is the Nusselt’s number and D is the machine diameter. The operating point has been selected as C p = 1.007 [KJ/KgK], μ = 19.2010–6 [Kg/sm], λ = 27.16*10–3 [W/mK]. Under the assumption of laminar motion where Pr·Gr is lower than 109 and turbulent for higher values, Nusselt equation parameters can be determined as follow: 102 < (Pr⋅ Gr) < 104 104 < (Pr⋅ Gr) < 109 10 < (Pr⋅ Gr) < 1012

C = 0.85, n = 0.188 C = 0.53, n = 1/4 C = 0.13, n = 1/3

After that, the convection coefficient h can be simply achieved considering a smooth motor outer surface, and it allows the defining of the outer surface boudary condition. h=

Nu ⋅ λ = 10 W/m2K D

Air-Gap Both convection and conduction within the air-gap are affected by the roughness of the rotor surface (K. Ball, B. Farouk e V. Dixit. [1]), the stator system and rotor rotational speed. In case of laminar motion, Reynolds number is small and the air-gap thermal conductivity coefficient is near to the still air. In case of turbulent motion, conduction coefficient is replaced by a new one that takes into account also convection transfer.

λ eff = 0, 0019 ⋅ η⋅−2,9084 ⋅ Re0,4614⋅ln(3,3361⋅η)

(7)


155

where η is shown in Eq. (8) η=

ro Ri

(8)

with 0.4 < η < 1. r0 is the rotor outer radius and Ri is the stator inner radius. Reynodls number is given by Re = r0 ⋅ ωm ⋅ δ / ν

(9)

where ωm is the rotor angular speed [rad/s], δ is the air-gap lenght [m] and ν kinetic viscosity [m2/s]. When smooth surface is assumed, the transition between laminar motion and turbulent motion is given by the Taylor’s number, ω2m rm δ3 ν2

Ta =

(10)

where rm =

δ is the average logarithmic radius ⎛r ⎞ ln ⎜ o ⎟ ⎝ ri ⎠

(11)

ro and ri are respectively the stator inner and outer radius. Following Becker and Koye theory, when Ta is lower than 1700 heat transfer is mainly devoted to laminar motion and Nusselt number is equal to 2, otherwise the following expression should be used:

h ⋅ 2g =2 λ N u = 0.128 ⋅ Ta 0.367 Nu =

N u = 0.41 ⋅ Ta 0.241

for

Ta < 1700 and g / ri → 0

for

1800 < Ta < 12000

for

12000 < Ta < 4 ⋅106

(12)

Air-gap heat transfer coefficient includes both conduction and convection and it Nu ⋅ λ can be written as h = . 2g Machine under test has 3 pole-pairs and it is fed with 50 Hz electrical, then Ω = 2π ⋅ f / p = 104.6 rad / s , rm = 67.5 m (5.14); g = 0.001 m; ν = 172.6 10–7 m2/s. Taylor number is equal to Ta = 2479, therefore turbulent motion in the air-gap determines the heat transfer which a Nusselt number equal to 2.25 and then h = 29.25 W/m2K. Concerning the air-gap, the data shown in [4] have been used, when the laminar motion and thermal conductivity coefficient is closer to quiescent free air.

156


Figure 5. Thermal performance of stator and rotor structure.

Figure 6. Thermal performance of stator structure.

Figure A1. Example of the structure under study.

Appendix In this section the comparison of the two proposed methodologies is shown. As example, it has been applied to the study of the thermal behavior of a simple structure shown in Fig. A1, where areas A1 and A3 are composed by iron (thermal conductivity λ Fe = 45 [W/mK] and size 0.03 × 0.03 [m]) and A2 area is air (adduction coefficient λ aria = 0.026 (T = 30°C) [W/mK] and size 0.03 × 0.03 [m]). Boundary condition are T1 = 40°C and T2 = 30 °C. At first, the air has been considered as a solid material with its thermal conductivity. After that, air has been considered as a fluid with its phisical characteristics. The achieved results have been compared. Once the materials, the boundary conditions and the mesh have been selected, thermal analysis has been carried out considering pure conduction heat transfer. The second step was the introduction of the thermal convection, fluid characteristics as well as the density temperature dependence. A comparison between the proposed methodologies is shown in Figs A.2 and A.3, where air temperature close to the borders is the same in both cases. Heat transfer is mainly due to thermal conduction being temperatures very close. Thermal convection


Figure A.2. Solution of the thermal analysis by means of the adduction coefficient.

157

Figure A.3. Air-iron temperature.

heat transfer can be taken into account by means of the adduction coefficient since the convection motion speed is quite low.

References [1] Christos Mademlis, Nikos Margaris, and Jannis Xypteras, Magnetic And Thermal Performance Of A Synchronous Motor Under Loss Minimization Control, Proc. Of the IEEE International Symposium On Industrial Electronics July 1995. [2] Y.K. Chin, D.A. Staton, Transient Thermal Analysis using both Lumped Circuit Approach and Finite Element Method of a Permanent Magnet Traction Motor, AfriCon 2004, pp. 1027–1035. [3] Ansys, Ansys Thermal Analysis Guide, November 2004. [4] J.M. Owen, Fluif Flow and Heat transfer in Rotating Disk Systems, Proc. Heat and Mass Transfer in Rotating Machinery, pp. 81–116, Springer Verlag, 1984.

158


Coordination of Surge Protective Devices Using “Spice” Student Version Carlos Antonio França SARTORI, Otávio Luís DE OLIVEIRA and José Roberto CARDOSO Escola Politécnica da Universidade de São Paulo, Departamento de Engenharia Elétrica PEA/EPUSP, Av. Prof. Luciano Gualberto Trav. 3, 158. O5508-900. São Paulo, SP. Brazil [email protected], [email protected], [email protected] Abstract. This paper presents a methodology concerning Surge Protective Device (SPD) Coordination Studies; taking into account the available tools of the student version of the “Spice”. As an approach to satisfy the related limitations of this version, preliminary analytical studies are carried out; allowing us to selected a list of the available Spice SPD models to be applied in the simulations. Some applications regarding residential installations were chosen, and their results are presented and compared with the ones presented in literature.

1. Introduction Nowadays, the electrical and electronic equipment and systems have been used in many branches of our modern society. On the other hand, the electromagnetic phenomena that they are exposed presents resulting effects which characteristics can be higher than their immunity levels, representing potential sources of Electromagnetic Interference (EMI). Concerning the electrical phenomena related to surges; it should be mentioned that suitable protection systems have to be designed. In particular, the adoption of Surge Protection Devices (SPD) is recommended, and the SPD Coordination studies are a project requirement: These studies have as objective to guarantee the suitable energy to be dissipated by SPD, besides the clamped voltages to satisfy the equipment immunity levels. Moreover, the improper usage of protective devices can result in several fails or damages to the electrical systems and equipment. In fact, the selection of those devices is not a simple task, and it requires many parameters to be considered like the kind of the surges waveform, its intensity and the associated energy, frequency of occurrence, the SPD configuration, the equipment immunity levels, etc. [1–5]. Based on the aforementioned scenario, it should be mentioned that the use of a single analytical methods can result in a complex and time-consuming work, and the application of computational tools appear as an interesting and a suitable approach. There are a lot of computational tools that can assist in solving electric circuits, but many of them are relatively expensive. For this reason, we have proposed, as part of the methodology, the use of a student version of a circuit simulator. Due to the wide use and literature availability, the so-called “Spice” was the software that the authors have chosen to be applied [6,7].

C.A.F. Sartori et al. / Coordination of Surge Protective Devices Using “Spice” Student Version

159

Figure 1. Methodology Flowchart.

2. Methodology Basically, the proposed methodology presents as main feature, preliminary analytical evaluations that allows obtaining pre-defined SPD models [3]. These SPD models will be, then, taken into account in the further Spice simulations. The details of the SPD, equipment models, and of the full method are briefly detailed in this section. Regarding the general aspects of the method, the following steps can describe it: 1. 2. 3. 4. 5. 6. 7. 8.

Definition of the surge characteristics according to Lightning Protection Zones (LPZ) [8]; Representation of the electrical system to be protected; Definition of the immunity level of equipment [9]; Definition of the SPD characteristics; Definition the pre-selected SPDs and configuration to be used in the computational simulation [3]; Implementation of the devices selected in the previous step in the “Spice” simulator; Computational simulation of the electrical system in study; Verification of the Surge Coordination.

The flowchart concerning the proposed methodology is presented in Fig. 1. 2.1. Definition of the Electromagnetic Environment The IEC 61312-3 standard presents the parameters that helps us to classify the electromagnetic environment of a structure to be protected [8]. These aforementioned areas or electromagnetic environments are called Lightning Protection Zone (LPZ). The Fig. 2 shows this principle concerning a pre-selected structure. It should be mentioned that the

160


Figure 2. Lightning Protection Zone division.

maximum value of voltage and current surges, the test wave-form, are defined according to the LPZ. 2.2. SPD Characteristics Regarding the SPD, it could be mentioned the following characteristics [2,3]: 1. 2. 3. 4. 5.

Under the rating voltage, the device will not conduct, although we can observe a small current, called leakage current, in this condition; For higher voltage, an electric current flows through the device, but the voltage across it will not increase significantly (Clamp Voltage); The energy capability of the SPD must be compatible with the surges energy level of the electrics systems; After the suppression of the surge, the SPD returns to the condition described in 1; The resulting clamp voltage must be smaller than the required immunity level of the equipment.

A typical device used in this study is the well-known varistor. 2.3. “Spice” Varistor Model The SPD models can be built based on the aforementioned characteristics and obtained directly from the available manufacturer technical literature. Regarding the varistor models, they can be considered as an association of 4 components, representing the physical phenomena that occur in the real varistor. Figure 3 presents an electric model of these devices. In the proposed varistor “Spice” model, initially, a series resistance (Rserie) is assumed, whose value is constant and equal to 100 nΩ. The components, Lserie and Cparalelo, represent series inductance and parallel capacitance of the device. These values change in accordance with the real model of the device. The variable re-


161

Figure 3. Spice varistor Model.

sistance (Rvariavel) is related to representation of the nonlinear characteristic of them. The dependence between voltage (v) and current (i) is simulated by a voltage generator, controlled by the current, and modeled by: log (V ) = p1 + p2 + p3 + p4

(1)

where, p1 = b1; p2 = b2 log(i); p3 = b3 exp (–log(i)); p4 = b4 exp (–log(i)). The parameters b1, b2, b3, b4 are related to specific characteristics of each varistor. 2.4. Details of the SPD Model Implementation in Spice Although the aspects concerning the software Spice have already been presented in various scientific papers, it should be emphasized some aspects of it. The Spice can be understood as formed by sub-routines, each one related to a specific part of the circuit simulation task. Firstly, it should be emphasized the structure and which files are used; in order to allow the model implementation in its library, like the varistor ones, in order to be available though the Schematics. In this case, the first file of interest is the “VAR.SLB” that contains the information regarding the graphical part of the devices: name, nodes, electrical parameters, mathematical characteristics, etc. This file should be associated to “PSpice” through the program Schematics, and it does not present any restrictions for the student versions. In fact, this file defines only the generic device of the component, called, e.g., “VAR”. The second file of interest is the “VAR.LIB”. This file has the mathematical definition of the characteristics of the component. It is where the parameters of each type of varistor (Cparalelo, Lserie, b1, b2, b3, and b4) are defined. This file should also be associated to “PSpice” through the program Schematics, and

162


SPD 1

SPD 2

Figure 4. Schematic representation of the circuits.

now the restrictions of the student version take an important role on this process. The Spice student version admits no more than 15 definitions of components in “VAR.LIB”. Preliminary analytical studies are proposed, as an approach to satisfy the related constrains of this version, allowing us to select a list of the available Spice SPD models to be applied in the simulations. Thus, the file “VAR.LIB” can be edited, and the non-selected device definitions deleted from the available list of components. 3. Application and Results Several results regarding different residential circuits were obtained. In order to validate the proposed SPD coordination approach, a configuration presented in literature was selected [4]: A residential one with eight branch circuits, which circuits were modeled as Transmission Lines (Z0 = 100 Ω, C = 50 pF/m and L = 0,5 µH/m). Figure 4 shows a schematic representation of circuits, and the corresponding Spice models. A surge waveform 20 kA (10 × 350 μs) was assumed in the simulations. Some results, related to different SPD arrangements, and energy dissipation, are presented in Table 1. Six different cases were simulated varying the configuration of the SPD, using four types of devices (S20K130, S20K150, S20K250 and S20K625). The energy dissipated on the SPD was compared with the device withstand energy, as well as the resulting clamp voltages were checked with the equipment immunity levels. Concerning the convergence simulation requirements, an analysis of sensitivity for the parameters named “ABSTOL”, “RELTOL” and “Step Ceiling” were carried out in order to opti-


163

Table 1. Resulting SPD energy SPD 1 Model

SPD 2 Model

Energy SPD 1 (J)

Energy SPD 2 (J)

0

-

1

20K625

-

-

-

-

23216

2

-

20K625

20K130

1255

496,261

3

20K150

-

1565,7

-

4

20K150

20K250

1467,2

89,566

5

20K150

20K130

1242,4

260,802

Configuration #

Table 2. Parameters used in simulations Case

Print Step Final Time Step Ceiling ABSTOL 1pA

RELTOL

0

0.1us

500us

0.1us

0.001

1

0.1us

4.5ms

0.1us

1pA

0.5

2

0.1us

1.6ms

0.1u

1pA

0.5

3

0.1us

3ms

0.1us

1pA

0.5

4

0.1us

3ms

0.1us

1pA

0.5

5

0.1us

3ms

0.01us

1pA

0.5

mize it. Notice that these parameters can affect the simulations, since they are directly related to approach used, like the Modified Node Analysis of the circuits [6,7]. The parameters used in each simulation are presented in Table 2.

4. Discussion Case 0 can be considered as a “control case”. It represents a residence without a surge protection system, and it is suitable for making comparisons with the cases in which the SPDs were adopted. Note that the results here presented are focused on the SPD energy dissipation. Cases 1 and 3 represent single SPD systems, whose SPD are positioned in the entry of the energy distribution system. The Case 2 counts with a second SPD, and it can be observed that the energy wasted in device S20K625 are smaller when compared to the Case 1 and Case 3. Case 4 presents the best alternative among the proposed protection configuration, presenting suitable energy coordination. The same conclusion can be observed in [4]. Case 5 does not present a good coordination between the SPDs. The Case 5 SPDs have a relatively low and close operation voltage rating, and the amount of energy dissipated in SPD 2 has increased when compared to the previous SPD configurations. Some differences on the results were observed when they are compared with the ones given in [4]: For example, the values of energy obtained in the Case 2 are smaller than the ones presented in this reference. On the other hand, in the Case 5, the energy observed on SPD 1 is lower, while the value of energy found for SPD 2 shows a value that is slightly above of the one presented in [4]. These differences can be attributed to the fact that was not possible to assume the same SPD characteristics in these works. As an example, the one of the devices used in the reference work that presents a nominal voltage equal to 200V was not available in the library of components that we have used, and it was substituted by another SPD (S20K130),

164


which Vn is 205 V. This can affect significantly the results due to the nonlinear behavior of those devices. That is, depending on the point of operation of the SPD, concerning its characteristic curve, small variations of terminal voltage can result in a wide difference in the resulting current. Besides that, as mentioned before, the parameters assumed to satisfy the convergence requirements, can also explain some of the differences on the results.

5. Conclusion An approach to satisfy the limitations of the Spice student version were presented based, preliminary, on analytical studies that allows us to selected a list of the available Spice SPD models to be applied in the simulations. Some applications regarding residential installations were chosen, and their results were here presented and discussed. Although some adequacies were adopted to satisfy the convergence constraints of the numeral method, the present methodology appears as a potential one to be used in the studies related to low voltage SPD coordination. It is emphasized the importance of this study, due to the relative low immunity levels of the equipments that is used in all branches of our society. An analysis of the influence of different load and circuit models are proposed as part of the future development of this work.

References [1] Lai, J.S.; Martzloff, F.D. “Coordinating Cascaded Surge Protection Devices: High-Low versus LowHigh”, IEEE Transaction on Industry Applications, Vol. 29, No. 4, pp. 680-687, July/August 1993. [2] Paul, D.; Srinivasa I.V. “Power Distribution System Equipment Overvoltage Protection”, IEEE Transaction on Industry Applications, Vol. 30, No. 5, pp. 1290-1297, September/October 1994. [3] Paul, D. “Light Rail Transit DC Traction Power System Surge Overvoltage Protection”, IEEE Transaction on Industry Applications, Vol. 38, No. 1, pp. 21-28, January/February 2002. [4] Standler, Ronald B. “Calculations of Lightning Surge Currents Inside Buildings”, 1992 IEEE International Symposium on EMC, Proceedings, pp. 195-199, Aug. 1992. [5] Standler, Ronald B. “Transient on the Mains in a Residential Environment” IEEE Transactions on Electromagnetic Compatibility, Vol. 31, No. 2, pp. 170-176, May 1989. [6] Tuinenga, P.W. “Spice: A guide to Circuit Simulation & Analysis using PSpice”, Prentice-Hall, 1988. [7] Herniter, Marc E. “Schematic Capture with PSpice”. Macmillan College Publishing Company, 1994. [8] IEC/TS 61312-3: 2000. International Electrotechnical Commission – “Protection against lightning electromagnetic impulse – Part 3: Requirements of surge protective devices (SPDs)”. [9] IEC 61000-4-5: 2001. Electromagnetic compatibility (EMC) of electrical and electronic equipment – Part 4: Testing and measurement techniques – Section 5: Surge Immunity test.

Chapter B. Computer Methods in Applied Electromagnetism B2. Numerical Models of Devices



167

Nonlinear Electromagnetic Transient Analysis of Special Transformers Marija CUNDEVA-BLAJER, Snezana CUNDEVA and Ljupco ARSOV Ss. Cyril & Methodius, Faculty of Electrical Engineering and Information Technologies, Karpos II b.b, POBox 574, R. Macedonia E-mail: [email protected] Abstract. In the paper original methodology for nonlinear electromagnetic transient analysis of special transformers will be given. A universal nonlinear transformer model will be developed by using the finite element method study results. The electromagnetic field analysis will be done by applying the original program package FEM-3D developed at the Faculty of Electrical Engineering and Information Technologies-Skopje (FEIT). The FEM results will experimentally be verified on a resistance welding transformer through actual test results recorded in a laboratory. The same methodology and model will be used for the metrological transient analysis of 20 kV combined current-voltage instrument transformer.

Introduction The special transformers, like resistance welding transformer (RWT) or combined current-voltage instrument transformer (CCVIT) are complex non-linear electromagnetic systems which operate in transient working regimes. The transients of the RWT are introduced by the nature of the welding process. The CCVIT must comply with the rigorous metrological specifications of the IEC 60044-2 standard [1] during the transient regimes. The RWT and CCVIT electromagnetic phenomena are described by the voltage equilibrium equations: 1 dψ1 ⋅ 1 1 ω dt b

u = i ⋅r + 1

u '2 = i '2 ⋅ r '2 +

1 dψ '2 ⋅ ωb dt

(1)

(2)

where, ψ 1 = ω b ⋅ λ1 , ψ 2 = ω b ⋅ λ 2 , λ1 and λ2 are the resultant fluxes created by the primary and secondary winding currents, respectively and ωb is the basic (industrial) frequency at which the reactances will be calculated by using the FEM. The RWT and CCVIT are non-linear bounded electromagnetic systems with prescribed boundary conditions and the electromagnetic field distribution is most suitably expressed by the system of non-linear partial differential equations of the Poisson’s type:

168

M. Cundeva-Blajer et al. / Nonlinear Electromagnetic Transient Analysis of Special Transformers

∂ ⎛ ∂A ⎞ ∂ ⎛ ∂A ⎞ ∂ ⎛ ∂A ⎞ ⎜ν B ⎟ + ⎜ν B ⎟ + ⎜ν B ⎟ = − j ( x, y , z ) ∂x ⎝ ∂x ⎠ ∂y ⎝ ∂y ⎠ ∂z ⎝ ∂z ⎠

( )

( )

( )

(3)

where the magnetic vector potential A as an auxiliary quantity is introduced, B is the magnetic flux density, ν the magnetic reluctivity and j ( x, y, z ) the volume current density. The analyzed devices are heterogeneous and (3) can be solved by numerical methods, only. The magnetic field analysis is done by an original and universal program package FEM-3D developed at the FEIT-Skopje, [2]. The results of the FEM analysis will be input data in a dynamic transformer model for transient analysis. The core saturation will be incorporated in the model as in [3] by using the relationship between measured saturated and unsaturated values of mutual flux.

Transformer Model for Transient Study The transient performance of transformers is influenced by a number of factors with most notable the exponentially decaying dc component of the primary current. Its presence influences the build-up of core flux, a phenomenon which is likely to cause saturation and subsequently substantial errors in the magnitude and phase angle of the generated signals. Core saturation mainly affects the value of the mutual inductance and, to a much lesser extent, the leakage inductances. Though small, the effects of saturation on the leakage reactances are rather complex and would require construction details of the transformer that are not generally available. In the FEM numerical approach transformer leakage inductance can be calculated by computing the normal flux at no load along a contour defined over the winding i.e. the normal leakage flux Φσ' . Multiplying the normal flux with the actual transformer length in the z direction lz yields:

Φσ = Φσ' ⋅ l z

(4)

Then the total leakage flux is defined as:

ψ σ = Φσ ⋅ N

(5)

where N are the number of turns excited. The leakage inductance is calculated by:

Lσ = ψ σ / I 0

(6)

where I0 is the no load current flowing through the primary winding. In many dynamic simulations, the effect of core saturation may be assumed to be confined to the mutual flux path. Different iron core models are described in the literature and their summary is described in [4]. In this paper a nonlinear SIMULINK transformer model has been developed and it is presented in Fig. 1. The effects of core saturation in the dynamic simulations have been incorporated using the relationship between saturated and unsaturated values of the mutual flux linkage as described in [3].

169


Initialize and plot

y To Workspace

Scope1

Scope

Plots FFT FFT

uslovi

Mux

Mux5

1

Clock

Out_psi1

Mux

wb*(u[2]-(r1/xl1)*(u[3]-u[1]))

s psi1_

Fcn

v1

Mux

psi1

1

i1

f(u)

3

Mux4

Scope4

Out_i1

Fcn4

Mux psim Mux

V

xM*(u[1]/xl1+u[2]/xpl2-u[3]/xm)

2

PQ I

Scope3 Active & Reactive Power

Mux3

Dpsi Memory1

v2p•

Mux

Out_psim

Fcn3

wb*(u[2] -(rp2/xpl2)*(u[1]-u[3]))

s psi2'_

Fcn2

Dpsi=f(psisat)

psi2'

1

Mux1

Mux

Mux2

(u[1]-u[2])/xpl2

i2'

Fcn5

4 Out_i2'

Load Module

Figure 1. SIMULINK transformer model for transient analysis. Table 1. FEMM numerical results

Φσ [Vs/turns]

ψσ [Vs]

Lσ [H]

magnetic field energy W [J]

5,656⋅10–6

531,6⋅10–6

118,13⋅10–6

1,877

normal leakage flux

total leakage flux

Leakage inductance

Magnetizing inductance Lm [H] 0,185

Study Case-RWT: Experimental Verification of the Model Prior to setting up a transformer model suitable for transient studies, a set of test results for commercial RWT were performed in the laboratory at FEIT. According to the manufacturer data and the measurements performed, the resistance welding transformer has the following rated data: primary voltage 380 V; secondary no-load voltage (1,41–4,63) V; conventional power 24 kVA; rated frequency 50 Hz; thyristor controlled switching; number of primary tap positions 9. The transformer is a single phase with shell type core. Core saturation has been determined from the open circuit magnetization curve of the investigated transformer. The leakage reactances of the RWT were calculated by using the FEMM program [5,6] and they served as input in the dynamic model shown in Fig. 1. The results from the numerical finite elements calculation are presented in Table 1. The derived SIMULINK results have been experimentally verified on the resistance welding transformer through actual test results recorded in a laboratory. A small selection of the derived results is presented in Table 2. The test results are satisfactory with the exception of a slightly different magnitude of the simulated iron losses Pfe and simulated reactive power Qfe. The simulation results for the currents are nearly identical to the laboratory tests. With these results the validity of the derived model has been proved. The transient performance of the RWT has been modeled by defining zero transition of the winding voltage β = 0. Sample results for the transient performance of the

170


Table 2. Comparison of the RWT experimental and simulated results Tap position 5 Experiment Simulation 0,65 0,80 55,4 58,0 78,3 83,0 115 100 217 285

I0 [A] noload current I1 [A] primary rms current I1max [A] primary magnitude current Pfe [W] active core losses Qfe [VA] reactive core losses

Tap position 8 Experiment Simulation 5,26 5,40 125 134 177 187 298 180 1529 2000

Figure 2. RWT switching transients versus time [s] at most rigorous phase angle β = 0.

analyzed RWT at nominal position 8 are shown in Fig. 2. From the results the exponentially decaying dc component of the primary current can be clearly observed. The current waveform displays large peak at the beginning (up to 100 times the rated value). Study Case-CCVIT: Application of the Verified Model The developed transformer model verified on the RWT study case is applied for transient analysis of the combined instrument transformer CCVIT with (voltage measurement core VMC ratio

20000 V

3

: 100 V

3

and current measurement core CMC ratio

100 A: 5 A). The CCVIT is with a complex electromagnetic construction and its geometry has been given in details in [7]. The electromagnetic parameters of the CCVIT are most important for its transient analysis, as they are the input data in the above developed SIMULINK transformer model. The CCVIT quasi-steady-state electromagnetic field analysis has been done by the original and universal program package FEM-3D, [2], developed at the Faculty of Electrical Engineering and Information Technologies in Skopje. The detailed CCVIT FEM-3D analysis has been given in [8]. The main electromagnetic characteristics of the CCVIT derived by FEM-3D, necessary for the transient analysis are given in Figs 3–8. In Fig. 3 the main flux characteristic per turn ϕmu in the upper middle cross-section of the VMC magnetic core via the VMC

M. Cundeva-Blajer et al. / Nonlinear Electromagnetic Transient Analysis of Special Transformers 40,0

30,0 20,0 10,0 0,0 0,0 0,5 1,0 1,5 Relative VMC input voltage U u /U ur [r. u.]

only VMC Ii/Iir=0 Ii/Iir=0,2 Ii/Iir=0,4 Ii/Iir=0,6 Ii/Iir=0,8 Ii/Iir=1,0 Ii/Iir=1,2

Figure 3. CCVIT main flux characteristics in [μVs] in the upper middle cross-section of the VMC magnetic core via the VMC input voltage and the CMC input current as a parameter.

Main flux per turn

Main flux per turn

40,0

30,0

only CMC Uu/Uur=0 Uu/Uur=0,2 Uu/Uur=0,4 Uu/Uur=0,6 Uu/Uur=0,8 Uu/Uur=1,0 Uu/Uur=1,2

20,0 10,0 0,0 0,0

1,0

1,5

Figure 4. CCVIT main flux characteristics in [μVs] in the upper middle cross-section of the CMC magnetic core via the CMC input current and the VMC input voltage as a parameter.

30

25 20

only VMC Ii/Iir=0 Ii/Iir=0,2 Ii/Iir=0,4 Ii/Iir=0,6 Ii/Iir=0,8 Ii/Iir=1,0 Ii/Iir=1,2

15 10 5 0 0,0

0,4

0,8

Leakage flux per turn

35 30

25 only VMC Ii/Iir=0 Ii/Iir=0,2 Ii/Iir=0,4 Ii/Iir=0,6 Ii/Iir=0,8 Ii/Iir=1,0 Ii/Iir=1,2

20 15 10 5 0

1,2

0,0

Relative VMC input voltage U u /U ur [r. u.]

0,0

0,4

0,8

1,2

0,8

1,2


Figure 6. VMC secondary winding (120 turns) leakage flux characteristics per turn in [μVs] via the VMC relative input voltage and the CMC relative input current as a parameter.


70 60 50 40 30 20 10 0

0,4

Relative VMC input voltage U u /U ur [r. u.]

Figure 5. VMC primary winding (24000 turns) leakage flux characteristics per turn in [μVs] via the VMC relative input voltage and the CMC relative input current as a parameter.


0,5

Relative CMC input current I i /I ir [r. u.]

40 Leakage flux per turn

171

25 20 15 10 5 0 0,0

0,4

0,8

1,2




Figure 7. CMC primary winding (6 turns) leakage flux characteristics per turn in [μVs] via the CMC relative input current and the VMC relative input voltage as a parameter.

Figure 8. CMC secondary winding (120 turns) leakage flux characteristics per turn in [μVs] via the CMC relative input current and the VMC relative input voltage as a parameter.

172


Figure 9. Switching transient primary CCVIT current of the VMC via time [s] at most rigorous phase angle β = 0.

Figure 10. Switching transient primary CCVIT current of the CMC via time [s] at most rigorous phase angle β = 0.

Figure 11. Time dependence of the VMC primary current (RMS value) at rated load of the both cores and β = 0.

Figure 12. Time dependence of the CMC primary current (RMS value) at rated load of the both cores and β = 0.

relative input voltage Uu/Uur and the CMC relative input current Ii/Iir as a parameter are displayed. In Fig. 4 the main flux characteristic per turn ϕmi in the upper middle crosssection of the CMC magnetic core via the CMC relative input current Ii/Iir and the VMC relative input voltage Uu/Uur as a parameter are displayed. The CCVIT transient analysis is done by coupling with the FEM-3D results. The magnetic field distribution results, e.g. leakage reactances characteristics are input data into the non-linear mathematical model of the CCVIT. The complex non-linear analysis has been done for rated loads of the both measurement cores and rated frequency of 50 Hz. The input voltage phase angle is β = 0. The CCVIT is a measurement device therefore its metrological parameters are of greatest interest. By using the SIMULINK transformer model the most important metrological CCVIT parameters have been calculated for the most rigorous moment the first forth of the signal period at the worst, from metrological point of view, regime at β = 0: the VMC voltage error pu = –17,5% and CMC current pi = –19%.


173

0,10

I1umax [A]

0,08

only VMC Ii/Iir=0.0 Ii/Iir=0.2 Ii/Iir=0.4 Ii/Iir=0.8 Ii/Iir=1.0 Ii/Iir=1.2

0,06 0,04 0,02 0,00 0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

Relative VMC primary voltage U u /U ur [r. u.]

Figure 13. Maximal VMC primary plug-in current via the VMC relative voltage, the relative CMC current is a parameter at β = 0 and rated loads of both cores.

120 100

I1imax [A]

80 60 40 20

only CM C Uu/Uur=0.0 Uu/Uur=0.2 Uu/Uur=0.4 Uu/Uur=0.8 Uu/Uur=1.0 Uu/Uur=1.2

0 0,0 0,5 1,0 1,5 Relative to rated steady- state regime CM C primary current I i /I ir [r. u.]

Figure 14. Maximal CMC primary plug-in current via the CMC relative current (through the CMC steadystate regime current), the relative VMC voltage is a parameter at β = 0 and rated loads of both cores.

Conclusions The developed transient performance transformer model has been verified as accurate on the resistance welding transformer and it has been further applied for transient analysis of a combined instrument transformer. The confirmed transformer model has been coupled with finite element method results. The methodology in the paper is universal and can be applied for other complex electromagnetic devices.

References [1] IEC (International Electrotechnical Commission) 60044-2, 1980: Instrument transformers, Part 3: Combined transformers, Geneve, 1980. [2] M. Cundev, L. Petkovska, The Weighted Residuals Method for Electromagnetic Field Problems in Electrical Machines, Proc. of the 32nd UPEC’97 Conference, Vol. 2, UMIST, pp. 934-937, 1997.

174


[3] C.M. Ong “Dynamic Simulation of Electric Machinery Using MATLAB/SIMULINK”, Prentice Hall PTR, Upper Saddle River, New Jersey, 1998. [4] Working Group C-5 of the Systems Protection Subcommittee of the IEEE Power System Relaying Committee, “Mathematical models for current, voltage, and coupling capacitor voltage transformers”, IEEE Transactions on power delivery, Vol. 15, No. 1, Jan. 2000, pp. 62-72. [5] D. Meeker, Finite Element Method Magnetics – User’s Manual 3.0, 1998-2000. [6] S. Cundeva, L. Petkovska., V. Filiposki., A Methodology for Coupled Steady-State ElectromagneticThermal Modeling of Resistance Welding Transformer, Proceedings of XI International Symposium on Electromagnetic Fields in Electrical Engineering ISEF’03, Maribor, Slovenia, 2003, pp. 751-756. [7] M. Cundeva, L. Arsov, G. Cvetkovski, Genetic Algorithm coupled with FEM-3D for metrological optimal design of combined current-voltage instrument transformer, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23, No. 3, 2004, pp. 670-676. [8] M. Cundeva-Blajer, L. Arsov, FEM-3D for Metrological Optimal Design and Transient Analysis of Combined Instrument Transformer, Przeglad Elektrotechniczny, R. 83 NR 7-8/2007, pp. 96-99.


175

Some Methods to Evaluate Leakage Inductances of a Claw Pole Machine a

Y. TAMTO a, A. FOGGIA a, J.-C. MIPO b and L. KOBYLANSKI b LEG, UMR5529 INPG/UJF-CNRS, BP46, 38402 Saint Martin d’Hères Cedex, France b VALEO Equipements Electriques & Moteurs, 2 Rue André Boulle-BP150, 94017 Créteil Cedex, France Abstract. To increase claw pole machine efficiency, we evaluate leakages in the magnetic circuit. These leakages are mainly due to end windings, air gap and slot leakages. Thus we obtain an equivalent circuit model. This model will be helpful to simulate the machine under load conditions and to make possible its optimization. The aim of this paper is then to present some methods of determination of leakage inductance, by computation and under test.

Introduction Methods for leakage inductance determination under test and computation, with the short circuit, the open-circuit and the zero-power-factor characteristic, to build the Potier diagram are well known. These methods were developed mainly for high power alternator. In the case of claw pole machine, zero-power-factor test and computation are difficult to run because of the value of inductances needed to simulate zero-powerfactor. This is why we explore other ways of leakage evaluation: we used the singular configuration of the claw pole alternator under load conditions, and the stator alone, without the rotor.

1. The Potier Method This model takes into account the magnetic saturation and is used in the special case of alternators with smooth poles. Figure 1 represents the Potier diagram in the general case. Under load, there is not phase difference between V and I because the load is a diode rectifier bridge charging a battery. We then compute λ with load characteristic, by solving Potier vector equations (1) and using finite element computation. Tests results will validate the parameters previously obtained. Ε 2r = (V + RI ) 2 + (ωλ I ) 2 I 2fr = I 2f + (α I ) 2 − 2α I cos(γ ) γ =

π λω I ) + arctg ( 2 V + RI

(1)

176

Y. Tamto et al. / Some Methods to Evaluate Leakage Inductances of a Claw Pole Machine

Er: Voltage behind leakage inductance λ : Leakage inductance α : Potier coefficient of equivalence Ifr: Load excitation corresponding to Er I f: Load excitation V: voltage /phase I: Current/phase R: Resistance/phase Figure1. Vector diagram of Potier.

2. Example The main characteristics of one of the claw pole machine used for tests are on the following table Winding Number of pole pairs Number of turns of rotor winding Rotor Resistance Number of phases Number of turns per phase Resistance per phase Number of stator slots Air-gap thickness

star 6 380 2,66 ohms 3 6 27 milliohms 36 0.365 mm

No load characteristic allow to express Ifr from Er, the parameter α is the slope of short circuit graph. Then with load tests we find λ by using electrical values at one phase boundaries at different speeds. Results for Potier parameters are: α = 0, 033 λ = 48μ H

The proposed method is more accurate for this type of alternator because test under load is more easy to carry out than the use of zero factor characteristic. But the computation time is long. So, we try to evaluate leakages by the removed rotor method.

3. Removed Rotor Method The rotor is removed and the stator is supplied by a three-phase current source, the amplitude of current varies from a very low value to the nominal one. The idea is that by removing the rotor, we take the main flux off; it only remains the leakage flux. The leakage inductance is then deduced from the electric quantities at the terminals of the stator windings. Another calculation of the leakages is done by using the electromagnetic energy stored.


177

Removed Rotor Test A tree phase current source is needed to feed stator phases. We made frequency and module variable. For each set of variables, the temperature of the windings is measured to evaluate the right resistance. Xa = V Z = I

Z

2

− R

2

(2)

Xa = ω λ: Leakage reactance; Z: Phase impedance; V: Voltage; I: Current; R: Resistance of the phase at T1 temperature R T1: = R T0. (1 + α.(T1 – T0)); α = 4 10–3 °C–1 for copper.

Removed rotor tests

Leakage inudctance (uH)

96

48

0 50

250

450

650

Freque ncy( Hz)

Figure 2. Leakage inductance.

Finite Element Computation The machine to computaee has three phases and 6 pole pairs. We choose the area under the stator teeth with tangent property for the magnetic field. A triplet of current is imposed in the phases [A, B, C] and they are distributed as follows: [IA, –IA/2, –IA/2]. Then leakage inductance is calculated by (3).

λ=

(4/ 3)*E Ι 2

A

E: electromagnetic energy. IA: current in phase A.

(3)

178


Figure 3. Design without Rotor.

Results IA (A) 5 10 20 30 40 50 60 70 80 90

Energy (J) 9,01E-04 3,61E-03 1,45E-02 3,26E-02 5,80E-02 9,07E-02 1,31E-01 1,78E-01 2,32E-01 2,94E-01

Leakage Inductance (μH) 48,1 48,2 48,3 48,3 48,4 48,4 48,4 48,4 48,4 48,4

We can notice that value of leakage doesn’t change with phase current and are the same for removed rotor computations and tests.

Conclusion The values of leakage inductances obtained by method exposed are nearly equals, this study validates removed rotor method for determination of leakage inductance and we can determinate alternator outputs with an accuracy of 99%. This method is speed and easy to test and to computate. In the aim of an optimization for example it’s useful to simplify the claw pole machine by its electrical model of Potier, this model allow to create a virtual test bench of the claw pole environment with numerical language.

References [1] IEEE Std 115-1995, pp. 50-66. [2] J. Dos Ghali, Essais Spéciaux sur les Machines Electriques.


179

Reduction of Cogging Torque in Permanent Magnet Motors Combining Rotor Design Techniques Andrej ČERNIGOJ a, Lovrenc GAŠPARIN a and Rastko FIŠER b Iskra Avtoeletrika d.d., Polje 15, 5290 Šempeter pri Gorici, Slovenia [email protected], [email protected] b University of Ljubljana, Faculty of Electrical Engineering, Tržaška cesta 25, 1000 Ljubljana, Slovenia [email protected] a

Abstract. High performance motor drive applications require permanent magnet synchronous motors (PMSM) that produce smooth torque with low torque ripple components. This paper quantifies various sources of torque ripple and is focused on rotor PMSM design techniques that can be used for reducing the cogging torque. For each chosen design technique a validation with finite element method (FEM) analysis is given. Finally the comparison and evaluation of design principles and their combination are presented and commented.

Introduction The ability of PMSM to produce smooth torque and high power density is important in high performance motor drive applications. Nevertheless, due to the mechanical construction and material properties there are various causes responsible for producing undesired torque ripple like: reluctance torque, cogging torque, and harmonics in the back emf. Several well-known design techniques [1–4] can be used to minimize this parasitic harmonic torque components, but most of them consequently reduce the output torque as well. Presented paper discusses and quantifies compromises among reducing cogging torque in connection with preserving the main output torque at high value in order to optimize the PMSM design for a specific drive application.

Instantaneous Torque of PMSM Instantaneous torque of a PMSM has two components: constant component T0 and periodic component Tr(α), which is a function of an electric angle α and presents torque pulsations called torque ripple (Fig. 1). T (α ) = T0 + Tr (α )

(1)

ˇ A. Cernigoj et al. / Reduction of Cogging Torque in Permanent Magnet Motors

180

Figure 1. The instantaneous torque of a PMSM motor T(α).

There are three origins of torque ripple in PMSMs: • • •

the difference between permeances in the air-gap in the d and q magnetic axis produces reluctance torque, cogging effect is the interaction between rotor magnetic flux produced by permanent magnets and variable permeance of the air-gap due to stator slots and produces cogging torque, distortion of sinusoidal distribution of the magnetic flux density in the air-gap produces field harmonic electromagnetic torque.

Rotor Design Techniques for Cogging Torque Reduction Elimination of cogging torque using available design techniques, while keeping the output torque at the same level is a challenging task. At the beginning the proper slot/pole combination must be selected for effective reduction of cogging torque. Beside this, for a given motor, the following rotor design techniques can be considered respectively or in any mutual combination: magnet span variation, magnet pole shifting, magnet shape and magnetization pattern optimisation, and magnet step skew or skewing. Because of the complexity of the given task, the full parametric FEM model of 36-slot and 6-pole PMSM was chosen. Due to slot/pole combination basic design will express a significant cogging torque, thus different rotor design techniques for cogging torque reduction can be studied and evaluated. Selection of Magnet Span with Magnet Pole Shifting Among several possibilities an effective way to reduce cogging torque, while maintaining the output torque, is to optimize the magnet span αm and shift magnetic pole angle γ, as shown in Fig. 2. To simplify the representation and enhance clearness only curves of maximal cogging torque Tcog max versus magnetic pole shift angle γ for several magnet spans αm are shown in Fig. 3. The minimal cogging torque values appear around the shift angle γ = 56° and are two to three times smaller than in the case of symmetrical magnetic pole distribution at γ = 60°. The influence of magnet span αm on cogging torque values can also be observed and it is obviously that the minimum appears at αm = 50°, thus optimizing of the PM motor design is an iterative process with many variable parameters.


181

Figure 2. Principle of magnetic poles shifting γ ≠ δ (left), and 2D FEM model of PM rotor (right).

Figure 3. Maximal cogging torque versus magnet pole shift angle γ.

Shape and Magnetization Pattern of Permanent Magnets Air-gap flux density distribution is strongly dictated by the shape and the magnetization pattern of applied permanent magnets. Furthermore this has a substantial influence on cogging torque, harmonic contents and magnetic saturation. Figure 4 shows various shapes and magnetization patterns of arc magnets usually used for surface mounted PMSMs. The influence of shape and magnetization pattern on the air-gap flux densitiy distribution is presented in Fig. 5. Notice that a bread loaf (lens-shaped) magnet shape flux densitiy distribution is the closest to the sinusoidal distribution. The analysis of shape and magnetization pattern of permanent magnets on cogging torque was carried out for all three shapes of PMs. Using a bread loaf magnet shape a minimal value of cogging torque compared to the constant component of the output torque T0 is achieved, but the output torque T0 is also considerable reduced in respect to the initial basic PMSP model, as presented in Fig. 6. Shifted magnet poles with selection of magnet span at the same time results in efficient reduction of cogging torque while keeping the level of the output torque T0 high (Table 1). Such approach represents more efficient solution in cogging torque minimization.

182


Figure 4. Various shapes and magnetization patterns of PM: surface radial (left), surface parallel (middle), and bread loaf (right).

Figure 5. Flux densitiy along the centre of the air-gap.

Figure 6. Maximal cogging torque and output torque for various shapes and magnetization patterns of PMs versus magnet span αm.

Conclusion Numerous parametric FEM calculations and laboratory measurements have proved that majority of rotor design techniques are very efficient in reducing the parasitic cogging


183

Table 1. Gathered results of rotor design techniques Design technique Basic PMSM model (αm = 56°) Optimal magnet span (αm = 40°) Magnet pole shifting with magnet span (γ = 56°, αm = 50°) Bread loaf magnet shape

Tcog max (Nm) 4,42 2,10 0,99 0,024

Tcog max / T0 17,7% 9,4% 4,0% 0,11%

T0 / T0 basic mod. 100% 89% 98% 89%

torque in PMSMs. If they are combined with several additional stator design techniques (optimization of slot openings, additional notches in stator teeth), the decreasing of the cogging torque Tcog max could be even improved, while maintaining the output torque T0 of the PMSM at the same level. Presented ascertainments are already brought into use in mass-production of PMSM motors for high demanding special applications.

References [1] J. Gieras, M. Wing, Permanent Magnet Motor Technology, New York, 1997, Marcel Dekker Incorporation. [2] N. Bianchi, S. Bolognani, Design techniques for reducing the cogging torque in surface-mounted PM motors, IEEE Trans. Industry Application, Vol. 38, No. 5, September/October 2002, pp. 1259–1265. [3] R. Lateb, N. Takorabet, F. M. Tabar, J. Enon, A. Sarribouete, Design technique for reducing the cogging torque in large surface mounted magnet motors, ICEM 2004 International Conference on Electrical Machines, Proceedings ICEM 2004 CD-ROM, Krakow, Poland, 5-8 Sept 2004. [4] M.S. Islam, S. Mir, T. Sebastian, Issues in Reducing the Cogging Torque of Mass-Produced PermanentMagnet Brushless CD Motor, IEEE Tran. Industry Application, Vol. 40, No. 3, May/June 2004, pp. 813820. [5] M. Furlan, A. Černigoj, M. Boltežar, A coupled electromagnetic-mechanical-acoustic model of a DC electric motor, Compel, Vol. 22, No. 4, 2003, pp. 1155-1165.

184


Optimum Design of Linear Motor for Weight Reduction Using Response Surface Methodology Do-Kwan HONG, Byung-Chul WOO, and Do-Hyun KANG Korea Electrotechonolgy Research Institute, P. O. Box 20, Changwon, 641-120, Korea Tel: +82-55-280-1395, Fax: +82-55-280-1547, E-mail: [email protected]

Abstract. This paper presents an optimum design procedure of linear motors to reduce the weight of the machines with the constraints of thrust and detent force using response surface methodology (RSM). RSM is well adopted to make the analytical model of the minimum weight with constraints of thrust and detent force, and it enables objective functions to be easily created, and a great deal of computation time can be saved. Therefore, it is expected that the proposed optimization procedure using RSM can be easily utilized to solve the optimization problem of the linear motors.

Introduction In many applications were the motion is essentially linear. It is possilble to use linear motors instead of rotary motors. Linear motors are electromagnetic devices developing mechanical thrust without mechanical slider-crank system mechanism. Advantages of the linear motors include low noise, reduced operating cost, and incearsed flexibility of operation due to gearless feature [1]. The linear motors, however, have some practical limitations. One of the major reasons of the limitations is that inherently large air gap causes low power density. In order to increase the power density, permanent magnet (PM) type longitudinal flux linear motors (LFLMs) can be considered for the application of the linear motors in high power density systems. Since LFLMs can produce high magnetic thrust and reluctance thrust with relatively small air gap. There are several practical examples of LFLM in [2,3]. In this paper we consider the development of a LFLM for use in linear compressor applications. Figure 1 shows the structure of the developed LFLM. It has two important electromechanical components, a linear actuator and springs for refrigeration application. For short stroke applications like in this situation in odrer to recover the energy at the end of the displacement mechanical springs are used. By controlling the operating frequency of the actuator around mechanical resonance frequency, the system has higher efficiency than conventional rotary type compressors. In order to increase the performance of the system it was mandatory to consider a method of optimization. RSM is recently receiving attention for its modeling ability of electromagnetic devices performance by using statistical fitting method, given that RSM is well adopted to develop analytical models for the complex problems. With this analytical model, an objective function with constraints can be easily created, and computation time can be saved.

D.-K. Hong et al. / Optimum Design of Linear Motor for Weight Reduction

185

Figure 1. PM type LFLM for compressor.

This paper presents the optimum design of longitudinal flux actuator for linear compressor using RSM. Its design goal is to reduce the weight of the machine with the constraints of thrust and detent force with respect to the initial machine. At first step, most influential design variables and their levels should be determined and be arranged in a table of orthogonal array. Each response value is determined by 3D finite element method (FEM). With the use of reduced gradient algorithm (RGA), finally, the most desired set is determined, and the influence of each design variables on the objective function can be obtained. The weight can be reduced by 10.09%, thrust force improved by 3.06% and detent force improved by 4.15% of initially designed PM type LFLM. Optimum Design for Longitudinal Flux Linear Motor (LFLM) RSM Method RSM seeks for the relationship between design variables and response through statistical fitting method. A polynomial approximation model is commonly used for a secondorder fitted response (u) and can be written as follow k

k

j =l

j =i

k

u = β 0 + ∑ β j x jj + ∑ β jj x j + ∑ β ij xi x j +ε 2

(1)

j =l

β : regression coefficients, x : design varaibles, ε : random error, k : number of design variables. The least squares method is used to estimate unknown coefficients. Matrix notations of the fitted coefficients and the fitted response model can be written as:

βˆ = ( X ′X ) −1 X ′u

uˆ = X βˆ

(2)

It should be evaluated at the data points. βˆ , where βˆ is the vector of the unknown coefficients which are usually estimated to minimize the sum of the squares of the error term. RSM method is applied in connection with FEM and the response actually represent FEM output values. Figure 2 presents the principal steps of RSM procedure.

186


Figure 2. RSM.

2.5

B (T)

2.0

S23

1.5 1.0 0.5 0.0

Laminated steel

0

10

20

30

H (kA/m)

40

50

60

Figure 3. B-H curves of the used materials. Table 1. Specifications of LFLM analysis model Item Stator/Iron material Permanent Magnet Nominal air gap Nominal current MMF

Unit

Specification

mm A

S23 NdFe35H 0.5 1 600

AT(Ampere Turn)

Design Variables and Levels Table 1 shows the specification of the PM type LFLM for a compressor. Figure 3 shows the B-H characteristic of the S23 material which was used for the active parts of the LFLM. The variables represent geometrical dimentsions which are completely determining the geometry of the LFLM no material variable was considered. Figure 4 shows the design variable of the PM type LFLM. Table 2 shows the design variable

187


Figure 4. Design variable of PM type LFLM. Table 2. Design variable and level Design Variable Level 1(–1) 2 (0) 3 (1)

DV1

DV2

DV3

DV4

DV5

DV6

DV7

8.449 9.94 11.431

5.95 7 8.05

1.9975 2.35 2.7025

4.25 5 5.75

5.95 7 8.05

1.833 3.666 5.499

16.15 19 21.85

and level. The table of orthogonal array is shown in Table 3 and corresponding simulation result (nominal current value 600 AT). The table of orthogonal array is determined by considering the number of design variables and each level of them. After obtaining the experimental data from 2D FEM, the function necessary to draw response surface is extracted. In order to determine the equations of the response surface, several experimental designs are developed to establish the approximate equation using the smallest number of experiments. The first level of DV6 and the second level of the other design varaibles have the values of the initially designed LFLM in Table 2. Optimum Design Result Table 2 represents the mixed orthogonal array, which is determined by considering the number of the design variables and each level of them. After getting the experimental data by 2D FEM, the function to draw response surface is extracted. In order to determine the equations of the response surface, several experimental designs are developed to establish the approximate equation using the smallest number of experiments. The purpose of this paper is to minimize the object function (weight) with constraints of thrust and detent force. Table 3 and Table 4 show optimum solution and comparison result of initial model and optimum model, respectively. The two fitted second order polynomial of the object functions for the seven design variables are as follows.

Weight = 1.44767 + 0.078 DV 1 + 0.07477 DV 2 + 0.01819 DV 3 + 0.05701DV 4 + 0.08289 DV 5 − 0.006 DV 6 + 0.00346 DV 7 − 0.00091DV 12 + 0.000465 DV 2 2 + 0.00196 DV 32 + 0.0024 DV 4 2 − 0.0006 DV 52 + 0.00335 DV 6 2 + 0.00129 DV 7 2 (3)

188


FThrust = 57.8049 + 4.6812 DV 1 + 3.2581DV 2 + 3.5414 DV 3 − 0.1728DV 4 + 0.0871DV 5 + 0.196 DV 6 − 0.5666 DV 7 + 0.0336 DV 12 − 0.0492 DV 22 2

2

2

2

− 0.2469 DV 3 − 0.1479 DV 4 + 0.0906 DV 5 + 0.2294 DV 6 − 0.0434 DV 7

(4) 2

FDetent = 0.38872 + 0.09841DV 1 + 0.03092 DV 2 + 0.07204 DV 3 − 0.03208DV 4 − 0.03093DV 5 + 0.12183DV 6 − 0.12523DV 7 + 0.05814 DV 12 − 0.03928 DV 42 (5) − 0.03498 DV 52 + 0.03957 DV 6 2 + 0.06569 DV 7 2

The adjusted coefficients of the multiple determination R2adj for three responses are weight (99.6%), FThrust (99.9%) and FDetent (98.1%). In Table 3 and 4, the optimal point is searched to find the point of less than 10.09% of the weight, 4.15% of detent force, and greater than 3.06% of the thrust force of the initially model. Table 4 and Table 5 show the optimum solution and the comparison result of initial model and optimum model, respectively. Figure 5 shows the 2D flux line and flux density of the initially model. Pareto chart of thrust force, detent force and weight are presented in Fig. 6. Also from this figure the most sensitive design variables can be identified. The response surface of thrust force according to change of the most influential design variables are shown in Fig. 7. Figure 8 shows an interaction plot of means for thrust force between design variables. Figure 9 shows a comparison between initial and optimum model. The weight can be reduced 10.09%, thrust force improved by 3.06% and detent force improved by 4.15% of the initially designed PM type LFLM. Figure 10 shows the detent force and thrust force profile of the optimum model by mmf. Table 3. Table of mixed orthogonal array L18(21×37) Exp.

DV1

DV2

DV3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

8.449 8.449 8.449 9.94 9.94 9.94 11.431 11.431 11.431 8.449 8.449 8.449 9.94 9.94 9.94 11.431 11.431 11.431

5.95 7 8.05 5.95 7 8.05 5.95 7 8.05 5.95 7 8.05 5.95 7 8.05 5.95 7 8.05

1.9975 2.35 2.7025 1.9975 2.35 3 2.35 2.7025 1.9975 2.7025 1.9975 2.35 2.35 2.7025 1.9975 2.7025 1.9975 2.35

DV4 DV5 DV6

DV7

4.25 5 5.75 5 5.75 4.25 4.25 5 5.75 5.75 4.25 5 5.75 4.25 5 5 5.75 4.25

16.15 19 21.85 21.85 16.15 19 21.85 16.15 19 16.15 19 21.85 19 21.85 16.15 19 21.85 16.15

5.95 7 8.05 7 3 5.95 8.05 5.95 7 7 8.05 5.95 5.95 7 8.05 8.05 5.95 7

1.833 3.666 5.499 5.499 1.833 3.666 3.666 5.499 1.833 3.666 5.499 1.833 5.499 1.833 3.666 1.833 3.666 5.499

Avg. thrust force(N) 46.72 53.23 59.41 50.60 58.30 64.42 58.80 66.76 61.65 53.27 49.92 55.72 54.53 60.47 57.85 62.72 57.81 66.71

Max.detent force(N) 0.33 0.33 0.4 0.37 0.34 0.46 0.39 0.97 0.36 0.53 0.38 0.25 0.46 0.33 0.49 0.43 0.33 0.91

Weight (kg) 1.16 1.37 1.62 1.36 1.6 1.41 1.49 1.46 1.66 1.37 1.37 1.37 1.35 1.43 1.59 1.57 1.49 1.54

Table 4. Optimum level and optimum size Design variable Level Optimum level Optimum size

DV1

DV2

DV3

DV4

DV5

DV6

DV7

0.508 10.70

–1 5.95

1 2.703

–1 4.25

–1 5.95

–1 0.4116 1.833 20.173


Table 5. Comparison of initial model and optimum model Model Initial Optimum (predicted RSM) 2D FEM (optimum simulation) Variation between initial and 2D FEM model (%)

Weight (kg) 1.4575 1.3099 1.3104

Avg. Thrust force (N) 57.8900 60.0000 59.6664

Peak. Detent force (N) 0.3594 0.3609 0.3445

–10.09

3.06

–4.15

(a) Flux line

(b) Flux density

Figure 5. Flux density and flux line in LFLM using FEM (initial model).

(a) Thrust force

(b) Weight

(c) Detent force Figure 6. Pareto chart of thrust force(Avg.), detent force and weight.

189

190


Figure 7. Surface plot of thrust force(Avg.).

Figure 8. Interaction plot of means for thrust force (avg).

0

0

600

Detent, thrust force (N)

60 50 40

Optimum (avg.) : 59.6664 N Initial (avg.) : 57.89 N

10

detent

120

thrust

100

initial

30 20

MMF(AT) :

600

optimum Initial (peak) : 0.3594 N Optimum (peak) : 0.3445 N

0

Detent, thrust force (N)

MMF (AT) :

0

300

600

900

1200

Optimum model

80 60 40 20 0

-3

-2

-1 0 1 Position (mm)

2

3

Figure 9. Comparison between initial and optimum model.

-3

-2

-1 0 1 Position (mm)

2

3

Figure 10. Detent and thrust force profile.


191

Conclusion In this paper, an optimum design procedure is introduced to design LFLM to reduce its weight and to improve thrust force and detent force of the initially designed LFLM with many shape design variables, Also, Optimization design by RSM and table of orthogonal array are presented in detail for the LFLM in this paper. The performance of optimized LFLM is improved as compared with the initial model. Based on this method the weight of optimized LFLM is reduced by 10%. Therefore, when this proposed approach is applied, it can efficiently raise the precision of optimization and reduce the number of iterations of experiments in the optimization design by RSM. References [1] H. Lee, S. S. Jeong, C. W. Lee and H. K. Lee, “Linear Compressor for Air-Conditioner,” International Compressor Engineering Conference at Purdue, pp. 1-7, 2004. [2] T. Mizuno, M. Kawai, F. Tsuchiya, M. Kosugi, and H. Yamada, “An Examination for Increasing the Motor Constant of a Cylindrical Moving Magnet-Type Linear Actuator,” IEEE Trans. Magn., Vol. 41, No. 10, pp. 3976-3978, October, 2005. [3] P. Zheng, A. Chen, P. Thelin, W. M. Arshad, and C. Sadaranani, “Research on a Tubular Longitudinal Flux PM Linear Generator Used for Free-Piston Energy Converter,” IEEE Trans. Magn., Vol. 43, No. 1, pp. 447-449, January, 2007.

192


Analytical Evaluation of Flux-Linkages and Electromotive Forces in Synchronous Machines Considering Slotting, Saliency and Saturation Effects Antonino DI GERLANDO, Gianmaria FOGLIA and Roberto PERINI Dipartimento di Elettrotecnica – Politecnico di Milano, Piazza Leonardo da Vinci, 32 – 20133 Milano, Italy [antonino.digerlando, gianmaria.foglia, roberto.perini]@etec.polimi.it Abstract. An analytical approach to the evaluation of flux linkages and e.m.f.s in salient-pole synchronous machines is developed, capable to accurately allow for the actual winding structure, the stator and rotor air-gap geometry (slotting and saliency), under any saturated, steady-state or transient operating condition. The method is based on a Park d-q decomposition of the stator m.m.f. distribution (preserving the spatial harmonics) and on the use of FEM identified saturation functions. A relevant feature is that the self and mutual inductances, evaluated in unsaturated conditions, are simply corrected by using the saturation functions. Several transient FEM simulations validate the method.

Introduction An analytical procedure for the evaluation of the air-gap field, of the inductances, of the e.m.f.s and of the electromagnetic torque of salient-pole, three-phase synchronous machines was previously developed, considering anisotropy, slotting and actual winding structure [1–3]: the procedure showed good accuracy features, but it was affected by the significant limitation of the operation in unsaturated conditions. Starting from that theory, the method is here extended to any saturated operating condition: this generalisation is based on a d-q approach to the calculation of the airgap field distribution, made possible by the use of FEM identified saturation functions [4] and by a d-q decomposition of the stator m.m.f. distribution [5]. In this paper, the expression of the coil and of the phase flux linkage and e.m.f. is obtained, while the electromagnetic torque evaluation is performed in another paper [6]. Based on the developed method, various simulations have been carried out, in different operating conditions, comparing the results with those obtained by corresponding transient FEM calculations [7]: the accuracy level is investigated and discussed, for the validation of the described analytical method. In the model, for now the presence of the rotor damper cage is not considered.

A. di Gerlando et al. / Analytical Evaluation of Flux-Linkages and Electromotive Forces

Stator phase global currents: i1s(t), i2s(t), i3s(t)

Park transformation on the rotating frame

Park current components:

Distinct antitransformation of the Park components

iPd(t) iPq(t)

193

Stator phase d, q current components: i1sd(t), i2sd(t), i3sd(t) i1sq(t), i2sq(t), i3sq(t)

Figure 1. Flow chart illustrating the d-q decomposition of the stator phase currents.

The d-q Decomposition of m.m.f. and Flux-Density Air-Gap Distributions As more completely described and illustrated in [5], by using the Park transformation, each stator phase current can be expressed in terms of phase d, q instantaneous components: i k ( t ) = i kd ( t ) + i kq ( t )

k = 1s, 2s, 3s.

(1)

The meaning of (1) is resumed by the flow-chart of Fig. 1. Thus, being ξ the generic angular position along the stator, the instantaneous m.m.f. distribution ms(ξ, i1s(t), i2s(t), i3s(t)) produced by the stator three-phase winding actual structure [1] can be decomposed as follows:

(

)

ms ( ξ,i1s ( t ) ,i 2s ( t ) ,i3s ( t ) ) = ms ( ξ,i1sd ,i 2sd ,i3sd ) + ms ξ,i1sq ,i2sq ,i3sq = = msd ( ξ, t ) + msq ( ξ, t ) .

(2)

As can be verified, the msd(t) and msq(t) distributions, produced by the d-axis and q-axis instantaneous current terns, act along the d, q axis respectively, showing the same stepped waveform features of the total m.m.f. [5]. Therefore, the model described by (1) and (2) represents a more general transformation than the classical Park one, usually applied to decompose sinusoidally distributed m.m.f.s only. As shown in [5], the instantaneous saturation functions are expressed by σd ( t ) = σd ( i r ( t ) ,id ( t ) ) ,

(

)

σq ( t ) = σq i u ( t ) ,iq ( t ) ,

(3)

where: id and iq are 1/ 3 times the current Park vector components iPd, iPq; ir is the rotor current; iu is an equivalent d-axis current corresponding to the resultant d-axis m.m.f. mu: i u ( t ) = m u N f = i r ( t ) + α Pd ⋅ id ( t ) ,

(4)

αPd is the classical d-axis Potier coefficient, Nf is the field winding turn number per pole. Called ζ = ζ(t) the rotor angular mechanical position, mr(ξ − ζ(t), ir(t)) the rotor m.m.f. distribution, βs and βr the stator slotting and rotor saliency functions respec-

194


tively [1], the air-gap flux density distribution (radial component, measured along the circumference at half the minimum air-gap width [5]) can be expressed as follows in saturated operation: b ( ξ, ζ ( t ) , t ) = ( μo g ) ⋅βs ( ξ ) ⋅βr ( ξ − ζ ( t ) ) ⋅

(

(

)

)

⋅ σd ( t ) ⋅ msd ( ξ, t ) + m r ( ξ − ζ ( t ) ,i r ( t ) ) + σq ( t ) ⋅ msq ( ξ, t ) = = σd ( t ) ⋅ bd.ns ( ξ, ζ ( t ) , t ) + σq ( t ) ⋅ bq.ns ( ξ, ζ ( t ) , t ) = = bd ( ξ, ζ ( t ) , t ) + bq ( ξ, ζ ( t ) , t ) .

(5) where bd.ns(ξ, ζ(t), t) and bq.ns(ξ, ζ(t), t) are the non-saturated d and q flux density field distributions. As can be observed in the first formulation of (5), the saturation functions are originally applied to the axis m.m.f. distributions: in fact, they can be interpreted as the factors that, along each axis, express the ratio between the distribution of the magnetic potential difference (m.p.d.) at the air-gap and the distribution of the m.m.f. The second line formulation of (5) shows that the d and q axis flux density distributions can be evaluated as the unsaturated ones, times the corresponding saturation functions.

The Development of the Flux-Linkage Expression in Saturated Conditions The flux linkage ψk of the phase k includes the main flux linkage (ψmk) and the leakage flux linkage (ψℓk): ψ k ( t ) = ψ mk ( t ) + ψ k ( t ) .

(6)

As regards the leakage flux linkage ψℓk, its rigorous analytical evaluation is difficult to be performed, as like as its FEM evaluation [3]. However, for the leakage model, here the well known assumptions of the classical theory will be adopted: − −

the leakage flux paths exhibit wide portions developing in air: thus, leakage can be considered as a substantially unsaturated phenomenon; the leakage flux linkage is assumed to be independent on the rotor position.

Considering the typical structure of the stator winding, in which the shorted coil pitch is usually employed, in general the following constructional properties occur: some slots contain conductors of different phases; moreover, some end-windings of different phases are close each others. As a consequence, the leakage flux linkage exhibits self and mutual terms, all due to the stator currents only; however, considering the three-phase structural symmetry, the phase self leakage inductances are all equal (Lℓself), as like as the phase mutual leakage inductances (Lℓmutual). Therefore, the leakage flux linkage of the phase 1s can be written as follows:


ψ 1s ( t ) = Lself ⋅ i1s ( t ) + Lmutual ⋅ i 2s ( t ) + Lmutual ⋅ i3s ( t ) = = ( Lself − 2 ⋅ Lmutual ) ⋅ i1s ( t ) = Ls ⋅ i1s ( t ) ,

195

(7)

where the “service” equivalent leakage inductance Lℓs has been introduced, thanks to the link: i1s ( t ) + i 2s ( t ) + i3s ( t ) = 0 ,

(8)

consequence of the hypothesis of the stator winding insulated neutral point. Of course, the leakage flux linkages of the other two phases have expressions similar to (7): ψℓk = Lℓs⋅ik, (k = 1s, 2s, 3s). As regards the main flux linkage ψmk, it can be expressed as a function of the teeth fluxes. From the expression (5) of the flux density distribution, the tooth flux in the j-th stator tooth [2] can be expressed as follows: ξfj b ( ξ, ζ, t ) ⋅ dξ = ξij

ϕ t j ( ζ, t ) = ⋅ R ⋅ ∫

⎛ ⎞ ξ ξ = ⋅ R ⋅ ⎜ σd ( t ) ⋅ ∫ fj bd.ns ( ξ, ζ, t ) ⋅ dξ + σq ( t ) ⋅ ∫ fj bq.ns ( ξ, ζ, t ) ⋅ dξ ⎟ = ξ ξ ij ij ⎝ ⎠ = σd ( t ) ⋅ ϕtd.ns j ( ζ, t ) + σq ( t ) ⋅ ϕtq.ns j ( ζ, t ) ,

(9) where ξij and ξfj are the j-th tooth initial and final angular positions (along the axes of the slots adjacent to the j-th tooth itself), ℓ the lamination stack length, R the average radius at the minimum air-gap (along the d axis). Equation (9) shows that also the tooth flux can be expressed as the sum of the unsaturated d and q axis components, ϕtd.nsj(ζ(t), t) and ϕtq.nsj(ζ(t), t), each multiplied by the corresponding saturation function. The main flux linkage ψmk of the phase k can be expressed as a function of the tooth fluxes: ψ mk ( t ) = Σ jΓ jk ⋅ ϕt j ( t ) ;

(10)

in (10) Γjk is an integer number, called linkage coefficient [2]: the value of Γjk expresses how many times the j-th tooth flux is linked with the k-th phase; the sign of Γjk depends on the winding direction of the phase winding coils, compared with the positive radial direction adopted for the generic tooth flux. Thanks to (9), the main phase flux linkage can be considered as the sum of the d and q components; moreover, each axis main flux linkage term can be expressed as the corresponding unsaturated main flux linkage component, multiplied by the suited saturation function; therefore, from (10) it follows: ψ mk ( t ) = ψ mdk ( t ) + ψ mqk ( t ) = σd ( t ) ⋅ ψ md.ns.k ( t ) + σq ( t ) ⋅ ψ mq.ns.k ( t ) . (11)

196


On the other hand, each non-saturated main flux linkage component (ψmd.ns.k(t) and ψmq.ns.k(t)) can be expressed as proportional to the corresponding current components (i1sd, i2sd, i3sd, ir and i1sq, i2sq, i3sq respectively), multiplied by the non saturated self and mutual inductance functions [2]: ψ md.ns.k ( t ) = L kr ( ζ ) ⋅ i r ( t ) + ∑ Lkv ( ζ ) ⋅ i vd ( t ); v

ψ mq.ns.k ( t ) = ∑ Lkv ( ζ ) ⋅ i vq ( t ); v, k = 1s, 2s, 3s.

(12)

v

The expressions of ψmd.ns.k(t) and ψmq.ns.k(t) of (12) can be considered as the limit of the main flux components ψmdk(t) and ψmqk(t) given by (11), when the saturation is negligible (i.e., when σd, σq → 1). Considering (7) and (12), (6) becomes (again with: v, k = 1s, 2s, 3s): ⎛ ⎞ ψ k ( t ) = σd ( t ) ⋅ ⎜ Lkr ( ζ ) ⋅ i r ( t ) + ∑ Lkv ( ζ ) ⋅ i vd ( t ) ⎟ + ⎜ ⎟ v ⎝ ⎠ ⎛ ⎞ + σq ( t ) ⋅ ⎜ ∑ L kv ( ζ ) ⋅ i vq ( t ) ⎟ + Ls ⋅ i k ( t ) . ⎜ ⎟ ⎝ v ⎠

(13)

The structure of (13) confirms the previous remarks concerning the possibility to express a saturated quantity as the product between the unsaturated one, times a saturation function; however, in this case, this property is particularly important, because it allows to evaluate the non saturated self and mutual inductance functions once and for all, as a function of the air-gap geometry and of the rotor position only, regardless of the current amplitude. Considering the heaviness of the inductances and inductance derivatives calculation process [2], the cited property is crucial for an acceptable application of the machine analytical model in saturated operating conditions. In order to illustrate the accuracy of the described flux linkage evaluation method, Fig. 2 shows the phase flux-linkage waveforms, in ideally unsaturated and saturated conditions, in steady-state, loaded, balanced operation, with d and q reaction components, for the 8-pole, fractional slot winding machine considered in [4] (waveforms evaluated analytically by (13) and by FEM transient simulations [7]). As can be seen, the calculation accuracy in saturated conditions is similar to that obtained in unsaturated ones, thus confirming the soundness of the saturation model: of course, the filtering effect of the winding distribution makes the waveforms almost sinusoidal. If the effect of saturation on more distorted quantities is to be examined, it is useful to analyse the coil flux linkage, shown in Fig. 3: in it, the waveforms correspond to those of Fig. 2, for the same machine and in the same loaded condition: as can be observed, the agreement between analytical and FEM results remains acceptable. The steady-state operating conditions of Figs 2 and 3 correspond to constant values σd and σq in (13). In order to evaluate the soundness of the analytical method in all the d and q axis saturation conditions, the most suited way is to consider an holding torque test; in this

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[Wb]

0.20

ψph.ns

4

ψc

0.50

0

0 −0.50

−2

−0.10

−4 −6

ψc.ns

0.10

ψph

2

[Wb]

0.15

−0.15 t [ms] 0

2

4

6

8

10

12

14

16

−0.20

18 20

Figure 2. Phase flux linkage at load (machine data: [4]): ψph.ns = unsaturated operation; ψph = saturated operation; solid lines = analytical; dotted lines = FEM.

0.1

4

6

ψc.ht

8

[ms]

10

12

14

16

18

20

ψph.ht.ns

[Wb]

4

ψph.ht

2

0

0

−0.1

−2

−0.2

−4

−0.3

−6 −8

−0.4

−10

−0.5 −0.6 0

2

Figure 3. Coil flux linkage waveforms; same machine and loaded operating conditions as in Fig. 2; solid lines = analytical results; dotted lines = FEM results.

6

ψc.ht.ns

[Wb]

t 0

−12

t [ms] 5

10

15

20

Figure 4. Coil flux linkage, in ideally unsaturated and in saturated operation, during rotation with constant ir, is1, is2, is3 values (holding torque test operation): solid line = analytical results; dotted line = FEM results.

−14

t [ms] 0

5

10

15

20

Figure 5. Phase flux linkage, in ideally unsaturated and in saturated operation, in the same conditions of Fig. 4 (holding torque test operation): solid line = analytical results; dotted line = FEM results.

condition, all the rotor and stator currents are maintained constant (I r, I1s, I2s, I3s) and the rotor is supposed to be driven at constant rotational speed Ω. So, the stator current Park vector has a constant amplitude, but its d, q components Id and Iq change with the rotor positions, in such a way to produce all the possible reaction situations (magnetizing (Id > 0), demagnetizing (Id < 0), generating (Iq < 0) and motoring (Iq > 0) conditions). Such operating condition should be considered as a virtual test, because difficult to be actually performed in case of large rating machines. The simulated results of this test are shown in Figs 4, 5, for the coil and phase winding flux linkage respectively: as can be observed, the correctness of (13) is confirmed. It should be noted the important flux linkage reduction due to saturation, that is unsymmetrical, depending on the magnetizing or demagnetizing reaction.

198


60

1.5

ec0.ns

[V] 40

[kV] 1.0

ec0

20

0.5

0

0

−20

−0.5

−40

−1.0

−60

t 0

5

10

eph0.ns

[ms] 15

20

Figure 6. Coil e.m.f. in no-load operation: ec0.ns = non saturated core; ec0 = saturated core; solid lines = analytical results; dotted lines = FEM results.

−1.5

eph0

t 0

5

[ms]

10

15

20

Figure 7. Phase e.m.f. in no-load operation: eph0.ns = non saturated core; eph0 = saturated core; solid lines = analytical results; dotted lines = FEM results.

The Development of the e.m.f. Expression in Saturated Conditions By performing the time derivative of (13), and posing Ω = dζ/dt, it follows:

(

)

e k = ekΩ + ekt + eks = Ω ⋅ σd ⋅ dLkr dζ ⋅ i r + ∑ v dLkv dζ ⋅ i vd +

(

)

+ Ω ⋅ σq ⋅ ∑ v dL kv dζ ⋅ i vq + σd ⋅ Lkr ⋅ di r dt + ∑ v Lkv ⋅ di vd dt +

+ σq ⋅ ∑ v L kv ⋅ di vq dt + Ls ⋅ di k dt + dσd dt ⋅ ∑ v ( Lkv ⋅ i vd + L kr ⋅ i r ) + + dσq dt ⋅ ∑ v L kv ⋅ i vq

v, k = 1s, 2s,3s.

(14) As can be noted, in addition to the two classical components (the speed e.m.f. ekΩ(t) and the transformer e.m.f. ekt(t)), a third term arises (that can be called “saturation” e.m.f.), eks(t), proportional to dσd/dt and to dσq/dt. It is interesting to analyze some particular operating conditions, again for coil and phase winding waveforms. Figures 6 and 7 refer to the steady-state, no-load, operation, with constant values of the speed Ω and of the rotor current Ir; in this condition, (14) reduces to the following no-load speed term: e k o = e kΩo = Ω ⋅ σd ( Ir , 0 ) ⋅ dL kr dζ ⋅ I r .

The following remarks can be made:

(15)

199


− −

as expected, the phase winding waveform is less distorted than the coil one, thanks to the filtering effect of the adopted winding structure (fractional slot, with shorted pitch coils); the waveform analytical reproduction is accurate, both in unsaturated and in saturated operation.

In steady state, loaded operation, the field current Ir is constant, while the stator currents i1s, i2s, i3s are balanced, sinusoidally time dependent quantities; considering that the d, q Park current components have constant values (Id, Iq), σd and σq can be evaluated once and for all; from (14), we obtain (again with v, k = 1s, 2s, 3s):

(

)

e k load = Ω ⋅ σd ( I r , Id ) ⋅ dLkr dζ ⋅ Ir + ∑ v dLkv dζ ⋅ i vd +

(

)

+ Ω ⋅ σq Iu , Iq ⋅ ∑ v dL kv dζ ⋅ i vq + σd ( Ir , Id ) ⋅ ∑ v ( Lkv ⋅ di vd dt ) +

(

)

+ σq Iu , Iq ⋅ ∑ v L kv ⋅ di vq dt + Ls ⋅ di k dt .

(16) Figures 8 and 9 show the coil and phase winding e.m.f.s respectively, in steady state, loaded, balanced operation, both in ideally unsaturated conditions and in saturated ones; the following remarks can be proposed: − − − − −

the saturation reduces significantly the waveforms amplitude, roughly maintaining their shape; a good agreement can be observed between analytically evaluated waveforms and FEM calculated ones, both in ideally unsaturated conditions and in saturated operation; the coil e.m.f. waveforms are highly distorted, because of the loaded operation; the phase winding e.m.f. waveforms are less distorted, thanks to the filtering effect due to the winding factor; however, the armature reaction affects also the distortion level of the phase e.m.f. waveform, mainly because of the third harmonic components; in fact, the line to line e.m.f. appears more sinusoidal.

The last operating condition here considered is the holding torque test, already examined in Figs 4 and 5 as concerns the coil and phase flux linkages; here, the coil and phase e.m.f.s are analyzed, both in ideally unsaturated operation and in saturated conditions. Considering that, in this case, the time derivatives of the currents are zero, (14) reduces to:

(

)

e k ht = ekΩht + eksht = Ω ⋅ σd ( Ir , Id ( t ) ) ⋅ dL kr dζ ⋅ Ir + ∑ v dL kv dζ ⋅ I vd +

(

)

+ Ω ⋅ σq I u ( t ) , Iq ( t ) ⋅ ∑ v dL kv dζ ⋅ i vq +

+ dσd dt ⋅ ∑ v ( L kv ⋅ I vd + L kr ⋅ I r ) + dσq dt ⋅ ∑ v L kv ⋅ I vq ; k, v = 1s, 2s,3s (17)

200


80

2.0

ec.ns

60 40

1.0

ec

0.5

20

[V]

0

0

−20

−0.5

−40

−1.0

−60

−1.5

−80

t 0

5

10

[ms]

15

20

−2.0

eph [kV]

t 0

5

10

3

[V]

ec.ht.ns

20

[kV]

2

50

eph.ht.ns

1 0

[ms]

15

Figure 9. Phase e.m.f. in loaded operation: eph.ns = non saturated core; eph = saturated core; solid lines = analytical results; dotted lines = FEM results.

Figure 8. Coil e.m.f. in loaded operation: ec.ns = non saturated core; ec = saturated core; solid lines = analytical results; dotted lines = FEM results.

100

eph.ns

1.5

eph.ht

0

ec.ht

−1

−50

−2 −100

−3 −150

t [ms] 0

5

10

15

20

Figure 10. Coil e.m.f., in ideally unsaturated and in saturated operation, during rotation with constant ir, is1, is2, is3 values (holding torque test operation): solid line = analytical results; dotted line = FEM results.

−4

0

5

10

t [ms] 15

20

Figure 11. Phase e.m.f., in ideally unsaturated and in saturated operation, in the same conditions of Fig. 9 (holding torque test operation): solid line = analytical results; dotted line = FEM results.

Figures 10 and 11 show the coil and phase winding e.m.f.s corresponding to the holding torque test, evaluated in ideally unsaturated conditions (i.e., when σd, σq → 1) and taking into account the saturation, on the basis of (17); the following remarks are valid: − −

the analytically evaluated waveforms are fairly similar to those evaluated by FEM transient simulations; the saturation has an important influence on the waveforms amplitude and shape;


−

201

the validity of (17) is substantially confirmed, even if some local discrepancies can be observed between analytical and FEM calculated waveforms, in the saturated operation.

Conclusion An analytical approach to the evaluation of the phase flux linkage and e.m.f. waveforms of salient-pole synchronous machines has been developed, able to take into account the actual structure of the stator winding, the slotting and anisotropy features under saturated operating conditions. The saturation has been modelled by using suited saturation functions, that are p.u. quantities, dependent on the amplitudes of the d-q m.m.f. main sinusoidal components: this saturation model allows to evaluate the actually saturated operating quantities as the corresponding unsaturated ones multiplied by the saturation functions. The most important consequence of this approach is that the self and mutual inductances, and the corresponding derivatives with respect to the rotor position, can be evaluated once and for all in unsaturated conditions, subsequently including the saturation effects corresponding to the actual conditions. Several operating situations have been analysed, comparing the waveforms calculated by the developed analytical approach with those obtained by means of corresponding transient FEM simulations: in general, the agreement appears satisfactory, showing the soundness of the developed method. References [1] A. Di Gerlando, G. Foglia, R. Perini: “Analytical Calculation of the Air-Gap Magnetic Field in SalientPole Three-Phase Synchronous Machines with Open Slots”, ISEF 2005 – XII Int. Symp. on Electromag. Fields in Mechatronics, Electrical and Electronic Eng., Baiona, Spain, Sept. 15-17, 2005, Proc. on CD, ISBN N° 84-609-7057-4, paper EE-3.14. [2] A. Di Gerlando, G. Foglia, R. Perini: “Calculation of Self and Mutual Inductances in Salient-Pole, ThreePhase Synchronous Machines with Open Slots”, ibidem, paper EE-3.15. [3] A. Di Gerlando, G. Foglia, R. Perini: “E.M.F. and Torque Analytical Calculation in Salient-Pole, ThreePhase Synchronous Machines with Open Slots”, ibidem, paper EE-3.16. [4] A. Di Gerlando, G. Foglia, R. Perini: “FEM identification of d-q Saturation Functions of Salient-Pole Synchronous Machines”, ISEF 2007 – XIII International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering, Prague, Czech Republic, September 13-15, 2007. [5] A. Di Gerlando, G. Foglia, R. Perini: “Analytical Model of the Air-Gap Magnetic Field in Synchronous Machines considering slotting, saliency and saturation effects”, ISEF 2007. [6] A. Di Gerlando, G. Foglia, R. Perini: “Analytical Calculation of the Electromagnetic Torque in Synchronous Machines considering slotting, saliency and saturation effects”, ISEF 2007. [7] Maxwell 2D FEM code, Version 10, Ansoft Corporation, Pittsburgh, PA, USA.

202


Radiation in Modeling of Induction Heating Systems Jerzy BARGLIK a, Michał CZERWIŃSKI b, Mieczysław HERING c and Marcin WESOŁOWSKI c a Silesian University of Technology, Krasińskiego 8, 40-019 Katowice, Poland E-mail: [email protected] b The Industrial Institute of Electronics, Długa 44/50, 00-241 Warsaw, Poland E-mail: [email protected] c Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland E-mail: [email protected], [email protected] Abstract. The paper presents analysis of third type boundary conditions applied for numerical simulation in more frequently used induction heating systems. A special emphasis was put on analysis on radiation heat transfer with taking into consideration multiple reflections phenomenon. Obtained results confirmed necessity of usage of multiple reflections model for analysis of high temperature induction heating systems.

Introduction Induction heating seems to be modern, environment-friendly industrial technology quite well explored from theoretical and practical point of view. Usage of the induction heating technologies leads to significant energy savings, distinct shortening the time of heating and consequently to the growth of a total efficiency. A powerful tool for designing and optimization is a suitable computer model based on a mathematical modeling of the task. The model is typically based on a system of non-linear second order partial differential equations for coupled electromagnetic and temperature fields. Development of the mathematical modeling of induction heating processes as well as the computations with required accuracy by means of professional software and sometimes also by some user codes has reached a high level. However there are some exceptions. One of them seem to be modeling of high temperature induction heating systems, for instance surface induction hardening of steel bodies. During such processes a radiation heat transfer plays an important role. The phenomenon is connected not only with high temperatures of the workpiece, but also with a typical arrangement of the system characterized by heated to high temperature charge and many surrounding elements with distinctly lower and often various temperatures. Majority of papers on induction heating present classical approach: taking into account radiation in a simplified way only without considering multiple reflections phenomenon. There are two reasons of the such the approach. One of them seems to be a false idea that multiple reflections do not influenced strongly on accuracy of calculations. The more important is the second reason: lack of professional software having precise, well done algorithms for such calculations. There are of course some packages having well prepared procedures for model-

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203

ing of radiation heat transfer with multiple reflections for instance TAS (Thermal Analysis System) but in that case software is not well prepared for electromagnetic calculations. The paper presents analysis of coupled electromagnetic and temperature in typical induction heating system with taking into consideration radiation heat transfer with multiple reflections.

Model of Radiation Heat Transfer in a System with Many Elements Let us consider an induction heating system with N surfaces of different temperatures T. For such a system the energy balance for each surface is given by Eq. (1), describing energy losses from the inductively heated charge to external surfaces [1]: N

⎛ δ k ,i 1− εi − ϕ k ,i ⋅ ε εi i =1 ⎝ i

∑⎜

⎞ Pi N 4 ⎟ ⋅ =∑ (δ k ,i − ϕk ,i )σ0Ti S ⎠ i i =1

(1)

where δ k ,i denote Kronecker delta defined as by (2): ⎧1 for k = 1 ⎫ δ k ,i = ⎨ ⎬ ⎩0 for k ≠ 1⎭

(2)

and ε i – effective emissivity of surface i, ϕ k ,i – view factor between two surfaces k and i, Pi – energy losses of surface i, σ 0 = 5.67·10–8 W/(m2·K4) – Stefan–Boltzmann constant, Si – area of surface i. Usually Si are elementary surfaces obtained which are lines or faces of a single elements of mesh. Main difficulty to solve the system of equations (1) is to calculate view factors ϕ k ,i , defined as the fraction of total radiant energy that leaves surface k which arrives directly on surface i. Another complication is that a radiating surfaces are a gray diffuse body, so total emissivity are less than one. Radiant energy that leaves surface k which arrives directly on surface i is not total absorbed on it, but according to Lambert law, is partly reflect and diffuse [1]. This effect of energy balance between all radiating surfaces in one enclosure called a reflection effect.

Formulation of Technical Problem and Illustrative Example Let us consider an axi-symmetric model of induction hardening system of cylindrical steel workpiece by three-turns cylindrical inductor. Main parameters of the induction heating system are as follows: Workpiece: diameter d = 100 mm; length l = 200 mm; thermal conductivity λ = 18 W/(m·K); density ρ = 8000 kg/m3; specific heat c = 500 J/(kg·K); effective emissivity ε = 0.8; conductivity γ = 7·106 S/m; relative magnetic permeability μr = 1;

204


Figure 1. Dependence of temperature of workpiece on time of heating with and without taking into consideration multiple reflections.

Inductor: number of turns n = 3; inner diameter Di = 130 mm; outer diameter De = 154 mm; width of one turn w = 16 mm; gap between turns Δw = 8 mm; effective emissivity of internal surface ε = 0.3 ÷ 0.8; Supply source: current density within the inductor J = 3·107 A/m2; frequency f = 1000 Hz. In order to determine influence of radiation heat transfer on accuracy of temperature calculations, in the first stage of computations a phenomenon of convection was neglected. Third type boundary condition having only radiation component was applied. We modeled the radiation heat transfer in induction heating process for several values of inductor temperature, its emissivity, distance between inductor and workpiece and shape of conductor. These elements have strongly influence on radiation heat transfer and on temperature profile in the workpiece. In order to simplify calculations steady-state analysis of temperature field was used. However before start the simulation it was necessary to solve also a transient problem. The obtained results (Fig. 1) show that maximal temperature differences are noticed in steady state. So in order to simplify the calculations in the further part of the paper only steady-state analysis will be provided. So the analysis could be done for that state. For calculations of weakly coupled electromagnetic and temperature fields Quick Field 2D (QF) and TAS packages were use. Electromagnetic calculations were made by means of QF (number of nodes is equal to 10428). Based upon results of specific power density released in the workpiece taken from QF steady-state temperature field

205


was calculated by means of TAS 3D. with 18373 nodes, 17800 elements, 17680 volume heat sources, 1720 radiation surfaces. Total power released in the workpiece is equal to 6702 W. Some results showing temperature distribution within the surface of the workpiece (Fig. 2) and in its longitudinal (Fig. 3) and transversal (Fig. 4) crosssections were presented below. O

C

Figure 2. Steady-state temperature field in workpiece heated by cylindrical inductor. O

Figure 3. Steady-state temperature field in a longitudinal cross-section of the workpiece.

C

206

J. Barglik et al. / Radiation in Modeling of Induction Heating Systems O

C

Figure 4. Steady-state temperature field in a longitidinal cross-section of the workpiece.

Temperature distribution in steady-state along length of the workpiece surface is presented in Fig. 5. Two cases were solved: with and without taking into consideration multiple reflection effect. The calculations confirmed that if multiple reflection phenomenon was neglected temperature of the workpiece is much lower (about 100 oC). The dependence between maximal temperature and total emissivity of the workpiece was shown in Fig. 6. For the case without multiple reflections (not shown in Fig. 6) maximal temperature is equal to 1197 oC. The calculations, that apply to inductor’s temperature influence on maximum temperature of charge proves, that he’s faint and can be entirely omitted. This is true in case of heating systems characteristic for hardening processes. In systems used in plastic forming of metals processes situation could be different. Assuming, that the workpiece transfer heat by natural convection, according to generally accepted formulas [2], the value of convection heat transfer coefficient changes for considered system and temperature range from α = 4.5 to 8 W/(m2·K), with practically constant value for temperatures bigger than 530 oC. Results of calculations are shown in Fig. 7. It follows from them, that participation of convection in heat transfer process is in comparison to radiation is rather small. It proves necessity to build model for temperature calculations taking into account radiation heat transfer with multiple reflections effect.

207


ε = 0.3

ε = 0.8

no reflections

1400 1300

Temperature, 0C

1200 1100 1000 900 800 700 600 0

25

50

75

100

125

150

175

200

Distance, mm Figure 5. Steady state temperature distribution on the workpiece surface of charge for two different values of emissivity and without multiple reflection phenomenon.

1300 1290

Temperature,0C

1280 1270 1260 1250 1240 1230 1220 1210 0.2

0.3

0.4

0.5

0.6

0.7

0.8

Emissivity, Figure 6. Maximum temperature of workpiece in function of total emissivity of inductor.

0.9

208


with convection

no convection

1400 1300

Temperature, 0C

1200 1100 1000 900 800 700 600 0

20

40

60

80

100

120

140

160

180

200

Distance, mm Figure 7. Steady state temperature on the surface of charge for inductor emissivity 0.3 with and without convection losses.

Modeling in ANSYS Environment Further simulations were made in ANSYS environment in 2D coordinate system. They include analysis of dependence of charges temperature in function of distance from inductor and its geometry. ANSYS is one of the rare packages, that allow simulation of radiation heat transfer with multiple reflection effect in both 2D and 3D coordinate systems. Such operation is impossible for TAS, that realizes this operation in 3D only. In purpose of objectivity in all cases the same volume power distribution in charge was maintained, as in earlier calculations. In purpose of comparison of simulation results obtained in different environments in Fig. 8, was shown temperature field received from ANSYS, with the same conditions and body loads, as presented previously (obtained with usage of QF+TAS packages). Figures 9 and 10 present temperature distribution on the surface of the workpiece for three different distances between inductor and the workpiece, with constant emissivity of inductor and workpiece and with the same power distribution within the workpiece. It follows, that value of emissivity has big influence on reflected radiation effect. This is frequently in classical models. On mentioned reflection effect essential influence has distance between inductor and workpiece L. The influence of the inductor can be omitted only for big values of distance inductor-workpiece L, that is avoided due to system efficiency.

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NODAL SOLUTION APR 27 2007 14:23:54

STEP=1 SUB =4 TIME=1 TEMP (AVG) RSYS=0 SMN =50 SMX =1287

MX

MN

50

421.156

235.578

606.733

792.311

977.889

1163

1287

Figure 8. Temperature field with 15mm distance of inductor from charge and emissivity 0.3.

no reflections

L = 15 mm

L = 35 mm

L = 55 mm

1300

Temperature, 0C

1200 1100 1000 900 800 700 600 0

20

40

60

80 100 120 140 160 180 200 220 Distance, mm

Figure 9. Temperature distribution on the charge surface of with constant emissivity ε = 0.8 and different distances of inductor from charge (L).

Last series of simulations concerned on influence of inductor geometry on temperature field distribution in the workpiece. Results are presented in Fig. 11. Change of the geometry in analyzed case was reduced to replacement of rectangular cross section of the conductor into circular one. It was assumed, that the surface of conductor cross section does not change and their geometrical centers are located in same place. The biggest influence of inductor geometry was observed with its small values of emissivity. In this case differences reached level of about 25 oC. For bigger values of emissivity this difference significantly decreased.

210


no reflections

L = 15 mm

L = 35 mm

L = 55 mm

1300

Temperature, 0C

1200 1100 1000 900 800 700 600 0

20

40

60

80 100 120 140 160 180 200 220 Distance, mm

Figure 10. Temperature distribution on the surface of charge with constant emissivity ε = 0.3 and different distance between inductor and charge (L).

circular turn ε = 0,8

circular turn ε = 0,3

rectangural turn ε = 0,8

rectangural turn ε = 0,3

1300

Temperature, 0C

1200 1100 1000 900 800 700 600 0

0.02

0.04

0.06

0.08 0.1 0.12 Distance, m

0.14

0.16

0.18

0.2

Figure 11. Temperature on the surface of workpiece in function of emissivity and for two different shapes of conductor.

Conclusions Obtained results confirmed necessity of consideration of multiple reflection effects in majority of induction heating systems. Errors caused by neglecting of multiple reflec-


211

tions phenomena may reach the order of 100 oC. For analyzed systems radiation takes a crucial part in heat transfer to surroundings. Convection is significantly smaller. Inductor is a kind of mirror, that reflect part of radiation and in result increasing temperature of charge (especially in close to surface area, that is most interesting in hardening processes).

Acknowledgement This work was financially supported by the Polish Ministry of Science and Higher Education (Grant Projects 9T08C 04678 and 503 /G/1041/0744/006).

References [1] Siegel R., Howell J.: Thermal radiation heat transfer. New York. McGraw-Hill 1972. [2] Hering M.: Termokinetyka dla elektryków. Warszawa. WNT 1980.

212


Time-Domain Analysis of Self-Complementary and Interleaved Log-Periodic Antennas A.X. LALAS, N.V. KANTARTZIS and T.D. TSIBOUKIS Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, GR-54124, Thessaloniki, Greece E-mail: [email protected] Abstract. A systematic near- and far-field analysis of dual polarisation logperiodic antennas is presented in this paper. The investigation of these demanding structures is performed by means of a 3D finite-difference time-domain (FDTD) methodology, properly tailored to tackle their geometrical details and abrupt discontinuities. In particular, the analysis delves into the design parameters of selfcomplementary, interleaved and trapezoidal-toothed structures and conducts a thorough examination of their radiation characteristics. Furthermore a new discretisation concept involving very flat cells which enhance algorithmic performance is introduced. Numerical verification addresses an extensive set of realistic applications with diverse parameter setups as well as instructive comparisons which indicate the merits of the proposed formulation.

Introduction Log-periodic antennas, introduced by DuHamel and Isbell [1], are typically used in wideband systems because their structure exhibits a practically independent behaviour with regard to operating frequency. As many applications require dual polarisation arrangements, potential choices comprise two different designs, i.e. the selfcomplementary (SC) and the interleaved (IL) configuration [2–5]. To this direction, one may also add an alternative design; the trapezoidal-toothed (TT) one, which shares similar attributes. Due to the inherent broad frequency spectrum, such antennas are essentially encountered in radar and measuring systems, attaining high-Q resonance properties. Amid the degrees of freedom of log-periodic antennas, one can discern their electrically thin ground-plane-backed dielectric substrate. So, prospective bandwidth advancement may be accomplished by increasing the substrate thickness or reducing its dielectric constant. However, the former issue is regularly accompanied by inductive impedance offsets and augmentation of the surface-wave effect. Thus, it becomes apparent that a meticulous procedure should be followed for the design of these particular radiators and the most significant: prior to any fabrication process to avoid expensive construction costs and totally misleading products. Towards the preceding deductions, the impact of design parameters on the radiation characteristics of log-periodic antennas has been comprehensively explored via several measurement attempts [6]. Moreover, various frequency-domain computational investigations have been presented, chiefly via commercial software packages [7]. Nevertheless, to our best knowledge, only a limited amount of time-domain attempts

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have been discussed. Therefore, it is the goal of this paper to introduce a FiniteDifference Time-Domain (FDTD) algorithm [8] for the efficient modelling and accurate simulation of 3D self-complementary, interleaved as well as trapezoidal-toothed antennas and derive reliable guidelines of their radiation characteristics. Special emphasis is drawn on the geometrical peculiarities of the devices along with their periodically-repeated attributes. Pursuing the improvement of the standard FDTD discretization rationale, a novel flexible method, incorporating very flat cells, is devised. In this manner, the domain is divided into more robust lattices devoid of erroneous mechanisms, oscillatory vector parasites or late-time instabilities. Additionally, open boundaries are truncated through diverse versions of the Perfectly Matched Layer (PML) absorber [9], which offer notable wave annihilation rates without any other non-physical conventions. To substantiate these qualities, the proposed approach is successfully applied to various real-world configurations, concerning all three structures, and numerical outcomes are carefully compared to extract possible similarities or prominent discrepancies. Structural Description of Log-Periodic Antennas The design parameters of a log-periodic antenna are α, β, Rmax, Rmin, the number of teeth N, geometrical ratio τ, and width ratio χ, as shown in Fig. 1. Specifically, τ and χ are given by τ=

Rn r = n −1 Rn +1 rn

and

χ=

rn Rn +1

(1)

and therefore, Rn+1 and rn are terms of geometrical progressions Rn +1 =

Rmin τ n +1

and

rn =

rmin τn

Figure 1. Geometry of a log-periodic planar element.

(2)

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(a)

(b)

(c)

Figure 2. Geometry of (a) a self-complementary, (b) an interleaved, and (c) a trapezoidal toothed structure.

During the design process, τ and χ can be acquired from τ =N

Rmin Rmax

and

χ= τ

(3)

given the rest of the parameters. Taking into account that the size of the antenna is fixed, we use (3) and then (2) to resolve the dimensions of metal surfaces. This is exactly the process followed throughout this paper. In particular, the self-complementary and interleaved structures are schematically depicted in Figs 2a and 2b, respectively. The antennas are realised through the use of two metallic sheets, properly shaped and mounted on a dielectric board, while the feed sections are constructed by integrated tapered-microstrip baluns. Two out of four branches are located on the upper side of the board and two are placed at its bottom. An alternative implementation of a dual polarisation device is the trapezoidal-toothed log-periodic radiator, whose geometry is illustrated in Fig. 2c. Accurate Analysis via the 3D FDTD Methodology The length and width of the antennas is set to 90 mm and their height is 6.67 mm. Rmax and Rmin are 40 mm and 5 mm, respectively, with these values kept fixed throughout the paper. The computational domain is divided into 144 × 144 × 44 cells with Δx = Δy = Δz = 0.8333 mm and Δt = 1.458 ps. Furthermore, termination of the unbounded space is attained by a 6-cell PML absorber. The two ports are excited by hard sources, appropriately phase-shifted to ensure circular polarisation. Our interest principally focuses on the variation of α, N and the relative dielectric permittivity εr of the dielectric board. From near and far-field data, acquired by our simulations, we can estimate the normalised (in dB) current distribution upon the surface of the structures and their radiation patterns. Some indicative results are illustrated in Figs 3a to 3c. It is emphasised that current values outside the outline of metal surfaces have no physical meaning, since they are merely a side-effect of the processing algorithm. As observed, maximum values of current distributions appear over the edges of the metal structures and hence the inner material is obsolete. This notification confirms previously-reported

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(a)

(b)

(c) ο

ο

Figure 3. Current distribution of (a) an SC antenna when N = 5, α = 80 , β = 10 , (b) an IL antenna when N = 4, α = 160ο, β = 10ο, and (c) a TT antenna when N = 5, α = 80ο, β = 10ο. 90

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Figure 4. Radiation pattern at x-plane (blue line for Eθ; red line for: Eφ) of (a) an SC antenna when N = 4, α = 80ο, β = 10ο, (b) an IL antenna when N = 5, α = 160ο, β = 10ο, and (c) a TT antenna when N = 5, α = 80ο, β = 10ο. 90

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Figure 5. Radiation pattern at z-plane (blue line for Eθ; red line for: Eφ) of (a) an SC antenna when N = 4, α = 80ο, β = 10ο, (b) an IL antenna when N = 5, α = 160ο, β = 10ο, and (c) a TT antenna when N = 5, α = 80ο, β = 10ο.

measurements [6] and leads to the wire structures. In addition, radiation patterns for each antenna, over the x- and z-plane, are presented in Figs 4 and 5, respectively.

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(a)

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Figure 6. Investigation (x-plane) on the variation of (a) N when α = 80 , β = 10 (blue line for N = 3; red line for N = 4, green line for N = 5; black line for N = 6), (b) α when N = 4, β = 10ο, (blue line for α = 20ο, red line for α = 40ο, green line for α = 60ο; black line for α = 80ο), and (c) εr when N = 5, α = 80ο, β = 10ο (blue line for εr = 2.8; red line for εr = 3.38, green line for εr = 4.4).

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Figure 7. Investigation (z-plane) on the variation of (a) N when α = 80 , β = 10 (blue line for N = 3; red line for N = 4, green line for N = 5; black line for N = 6), (b) α when N = 4, β = 10ο (blue line for α = 20ο; red line for α = 40ο, green line for α = 60ο; black line for α = 80ο), and (c) εr when N = 5, α = 80ο, β = 10ο (blue line for εr = 2.8; red line for εr = 3.38, green line for εr = 4.4).

The impact of N, α variation and that of the relative dielectric permittivity εr on the radiation characteristics is next examined. Radiation patterns for an SC structure, over the x- and z-plane, are shown in Figs 6 and 7. More specifically in Figs 6a and 7a, an investigation on the behaviour of N is provided. Because of the fixed size of the antenna, any change of N is actually translated to a corresponding change of τ and therefore teeth dimensions vary when a supplementary one is inserted. As a result, teeth resonate in different frequencies for each case but the overall performance of the antenna is similar for small modifications. On the other hand, Figs 6b and 7b give an investigation on the variation of α. As α increases, a more bidirectional behaviour is revealed. This is because for smaller angles, dimensions of the teeth are smaller too and thus can not resonate. Subsequently in Figs 6c and 7c, an examination on the effect of relative dielectric permittivity εr is depicted. The attitude of the antenna is, now, more bidirectional for smaller εr. In this context, for a substrate with a high dielectric constant, the effective length of the teeth

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(c)

Figure 8. Comparison of radiation patterns (x-plane) for different (a) frequencies, when N = 5, α = 80ο, β = 10ο (blue line for f = 2.675 GHz; red line for f = 2.867 GHz, green line for f = 3.6352 GHz), (b) structures (blue line for SC; red line for IL, green line for TT), and (c) distances of metallic surfaces at 2.0028 GHz, when N = 5, α = 80ο, β = 10ο (blue line for Δz = 0.8333 mm; red line for Δz = 0.2525 mm, green line for Δz = 0.1263 mm).

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Figure 9. Comparison of radiation patterns (z-plane) for different (a) frequencies, when N = 5, α = 80ο, β = 10ο (blue line for f = 2.675 GHz; red line for f = 2.867 GHz, green line for f = 3.6352 GHz), (b) structures (blue line for SC; red line for IL, green line for TT), and (c) distances of metallic surfaces at 2.0028 GHz, when N = 5, α = 80ο, β = 10ο (blue line for Δz = 0.8333 mm; red line for Δz = 0.2525 mm, green line for Δz = 0.1263 mm).

is decreased. Thus, when increasing the value of εr, lower frequencies can not resonate [10,11]. Next, some additional comparisons are conducted. Radiation patterns of an SC structure, for different frequencies, are shown in Figs 8a and 9a. Bi-directional behaviour is observed in each case. Moreover, in Figs 8b and 9b a comparison of the structures presented earlier is depicted. Their behaviour is satisfactory in the sense of being bi-directional. The differences on the amplitude of the radiation patterns are explained taking into account the mismatching of the ports. In order to avoid them a more accurately modelling of the ports is needed. It is noteworthy to observe the performance of the antenna in the case of reducing the distance between its two metallic surfaces. To model these intricate cases, a new flat-cell discretization approach, namely Δx = Δy >> Δz, is developed. We examine two cases. In the first, the height of the antenna is 2.0202 mm and the FDTD domain is divided into 148 × 148 × 90 cells with Δx = Δy = 0.8333 mm, Δz = 0.2525 mm,

218


and Δt = 0.703 ps. In the second case, the height of the antenna is 1.0101 mm. The FDTD domain is divided into 148 × 148 × 146 cells with Δx = Δy = 0.8333 mm, Δz = 0.1263 mm, and Δt = 0.374 ps. Again, termination of open boundaries is attained by a 6-cell PML along x- and y-axis and a larger PML setup towards z-axis. This is deemed necessary to ensure that the thickness of the absorber will be the same at every direction in the grid. As we can notice from Figs 8c and 9c when the distance between the two metallic surfaces decreases, the behaviour of the antenna at a specific frequency follows a more omni-directional pattern, basically due to loss of antenna resonance. Moreover, lower frequencies are suppressed at the entrance of the tapered-microstrip baluns owing to the smaller height of these waveguiding structures. Consequently, they can not excite the metallic parts of the antenna. Overall, the preceding investigations prove that bandwidth enhancement may be conducted either by augmenting the thickness of the substrate or by decreasing its dielectric constant.

Conclusion A consistent 3D FDTD technique for the rigorous analysis of log-periodic antennas has been presented in this paper. Three different structures with respect to their periodical features are investigated, whereas for improved precision a new flat-cell approach is developed. Numerical validation involves a variety of simulations and comparisons between different antenna types. Particularly, the influence of the teeth number and spanning angle on the antenna overall performance are extensively explored and combined with a study on the effect of the relative dielectric constant and the distance of the basic metallic surfaces. Results confirm the benefits of the proposed time-domain method and demonstrate the potential to be employed as a promising tool for logperiodic antenna characterization. Future aspects involve a more detailed modelling of the ports in order to obtain higher levels of accuracy.

Acknowledgement This work was supported by the National Scolarships Foundation of Greece (IKY).

References [1] R.H. DuHamel, and D.E. Isbell, Broadband logarithmically periodic structures, Record of 1957 IRE National Convention, Part 1, 99, pp. 119-128. [2] A.L. Van Hoozen, et al., Conformal log-periodic antenna assembly, US Patent 6,011,522, 4 Jan. 2000. [3] A.L. Van Hoozen, et al., Bidirectional broadband log-periodic antenna assembly, US Patent 6,018,323, 25 Jan. 2000. [4] D.A. Hofer, et al., Compact multipolarised broadband antenna, US Patent 5,212,494, 18 May 1993. [5] D. Campbell, Polarised planar log-periodic antenna, US Patent 6,211,839 B1, 3 April 2001. [6] R.H. DuHamel, and F.R. Ore, Logarithmically periodic antenna designs, Record of 1958 IRE National Convention, Part 1, Vol. 100, pp. 139-151. [7] K.M.P. Aghdam, R. Faraji-Dana, and J. Rashed-Mohassel, Compact dual-polarisation planar logperiodic antennas with integrated feed circuit, IET Microw. Antennas Propag., Vol. 152, pp. 359-366, 2005. [8] A. Taflove, and S. Hagness, (3rd ed.), Computational Electrodynamics: The Finite-Difference TimeDomain Method, Boston: Artech House, 2005.


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[9] J.P. Berenger, Three-dimensional perfectly matched layer for the absorption of electromagnetic wave, J. Comp. Phys., Vol. 127, pp. 363-379, 1996. [10] E. Avila-Navarro, J.M. Blanes, J.A. Carrasco, C. Reig, and E.A. Navarro, A new bi-faced log periodic printed antenna, Microw. Optical Technol. Lett., Vol. 48, pp. 402-405, 2006. [11] B.L. Ooi, K. Chew, and M.S. Leong, Log-periodic slot antenna array, Microw. Optical Technol. Lett., Vol. 25, pp. 24-27, 2000.

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New Spherical Resonant Actuator Y. HASEGAWA a, T. YAMAMOTO a, K. HIRATA a, Y. MITSUTAKE b and T. OTA b a Department of Adaptive Machine Systems, Osaka University, Yamadaoka, 2-1, Suita-city, Osaka 565-0871, Japan [email protected] b Advanced Technologies Development Laboratory, Matsushita Electric Works, Ltd., 1048, Kadoma, Osaka 571-8686, Japan [email protected] Abstract. This paper proposes the new spherical resonant actuator. The basic construction and the operating principle of the actuator are described. The torque characteristics of the actuator are computed by the 3-D FEM analysis. The geometry of the mover is investigated to improve the torque characteristics and the effectiness is clarified by both of the computation and the measurement of a prototype. Futhermore, the dynamic characteristics of the improved model are also confirmed by the measurement.

1. Introduction Recently, multi-dimensional actuators are a topic of great interest because of solution for vibration, noise, size constraints and limitations on operating speed [1]. Particularly, spherical actuators [2] are studied as the application to the joints and eyeballs for robots because they can be freely rotated in every axis direction. Due to the computer progress, the computer simulation becomes an effective tool to design electric devices and actuators. Authors have been studying the analyzed method employing the 3-D FEM to apply to multi-dimensional actuators with complicated magnetic structure [3]. In this paper, the new spherical actuator is proposed and the torque charactersistics are computed through the FEM analysis. The validity of the computation is verified by the comparison with the measurement of a prototype. The resonance characteristics around two rotation axes are confirmed through the measurement.

2. Basic Structure and Operating Principle Figure 1 shows the basic construction of the proposed spherical resonant actuator. It has the hybrid magentic structure [4] so that the magnetic flux by the current can not flow through the permanent magnet because permanent magnet has large magnetic resistance. The mover has four magnetic poles made of cross-shaped iron, permanent magnets (Br = 1.42 T), and spherical iron cores. The stator has four spherical magnetic poles with exciting coils of 100 turns. The air-gap between both spherical faces is

Y. Hasegawa et al. / New Spherical Resonant Actuator

221

Figure 1. Basic construction of the proposed model.

Figure 2. Magnetic circuit of the basic model.

Figure 3. Torque characteristics of the basic model.

0.3 mm. The mover is connected to four common resonance springs to be operated in resonance frequency. Figure 2 shows the cross section of x-z plane of the basic model. The magnetic flux by permanent magnet flows along the solid line, and the mover keeps balance at the center. When the coils are excited as shown in this figure, the flux flows along the dotted line, and the flux in the air-gap becomes unbalanced, and torque is generated. The mover can be rotated around arbitrary axis by changing the amplitude and direction of four coil currents.

3. Static Torque of the Basic Model Figure 3 shows the computed torque characteristics of the basic model employing the 3-D FEM when the coil A and C are excited, and the mover is rotated from 0 to 5 degree in step of 1 degree around y-axis. The average torque constant is 13.9 × 10–2 mN·m/A. Figure 4 shows the distribution of magnetic flux vectors with the coil excitation of 0 and 100 A. When coils are not excited, the magnetic flux by the magnet flows around the mover ploes and the upper parts of stator poles. On the other

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(a) 0AT

(b) 100AT (Coil A + Coil C) Figure 4. Distribution of flux density vectors.

hand, when both of coil A and coil B are excited, the magnetic flux by the current flows through the yoke of whole stator and the mover. It is found that the magnetic path by the current is completely divided with the magnetic path by the magnet.

4. Static Torque of Improved Model Figure 5 shows the improved model, which has the same structure as the basic model mentioned above except the mover. The mover of this model has the full-scale spherical yoke in order to keep the facing area between the mover and the stator, and has the ring-shaped permanent magnet (Br = 1.42 T) at the center. Figure 6 shows the comparison between the measured and the calculated torque characteristics with the coil excitation of 0 and 100 A. As can be seen, both results are in good agreement. This actuator has the stable position at the rotation angle of 0.0 degree. The computed average torque constant is 27.4 × 10–2 mN·m/A. It is twice as large as the basic model. Figure 7 shows the distribution of the flux density vectors, Fig. 8 shows the contours of the flux density in the neighborhood of facing area. The magnetic flux density of the improved model becomes stronger than that of the basic model, because the magnetic flux from whole of the ring-shaped magnet flows into the stator poles. As a result, the average torque constant becomes large. And the cogging torque characteristic of the improved model shows the linearity versus rotation angle compared with the basic model.

5. Dynamic Characteristics of Improved Model Figure 9 shows the prototype of the proposed model, which has the gimbal mechanism to be operated in the spherical surface with 0.3 mm gap between mover and stator, and has four resonant springs on the upside of the mover. Figures 10 and 11 show the measured frequency characteristics for x- and y- directions while it is operated at the same time. When the input voltage of 2.4 V (peak to peak) is applied with resonance frequency of 173 Hz for x-direction, the, the maximum rotation angle is 4.3 degree (peak


223

(a) overall view

(b) x-z section Figure 5. Construction of the improved model.

to peak) and the average current is 0.4 A. On the other hand, when the same voltage is applied with resonance frequency of 127 Hz for y-direction, the maximum rotation

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Figure 6. Comparison between measured and calculated torque characteristics of improved model.

(a) Basic model

(b) Improved model Figure 7. Distribution of flux density vectors.

angle is 6.4 degree (peak to peak) and the average current is 0.5 A. The difference of resonance frequencies for x- and y- directions is due to the inertia of the gimbal mechanism. Figure 12 shows the trajectory of the mover when it is operated at resonant frequencies of x- and y-directions (173 Hz and 127 Hz). As shown, it is found that this actuator can be operated in arbitrary direction.


(a) Basic model

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(b) Improved model

Figure 8. Contours of flux density in the neighborhood of facing area.

Figure 9. Prototype.

6. Conclusions This paper presentedthe new spherical resonant actuator. The torque characteristics were computed through the 3-D FEM analysis. The effect of mover geometry on the torque characteristics was investigated. And, it was found that average torque of the improved model was twice as large as the basic model. The validity of the computation was verified by the comparison with the measurement of a prototype. Furthermore, the dynamic characteristics versus rotation angle were confirmed through the measurement. As a result, it was found that the proposed actuator was operated in spherical surface. This research was supported in part by “Special Coordination Funds for Promoting Science and Technology: Yuragi Project” of the Ministry of Education, Culture, Sports, Science and Technology, Japan.

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Figure 10. Measured frequency characteristics (x-direction).

Figure 11. Measured frequency characteristics (y-direction).

Figure 12. Trajectory of multi-motion. (x direction: 173 Hz, y direction: 127 Hz).

References [1] A. Tanaka, M. Watada, S. Torii and D. Ebihara, “Proposal and Design of Multi-Degree-of-Freedom Spherical Actuator”, 11th MAGDA Conference, PS2-3, pp. 169-172, 2002. [2] E.h.M. Weck, T. Reinartz, G. Henneberger and R.W. De Doncker, “Design of a spherical motor with three degrees of freedom”, Annals of the CIRP, Vol. 49, pp. 289-294, 2000.


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[3] K. Hirata, T. Yamamoto, T. Yamaguchi, Y. Kawase and Y. Hasegawa, “Dynamic Analysis Method of Two-Dimensional Linear Oscillatory Actuator Employing Finite Element Method”, IEEE Transaction on Magnetics, Vol. 43, No. 4, pp.1441-1444, 2007. [4] K. Hirata, Y. Ichii and Y. Kawase, “Novel Electromagnetic Structure with Bypass Magnetic Path for Reset Switch”, IEEJ Trans. IA, Vol. 125, No. 3, pp. 293-296, 2005.


Chapter C. Applications C1. Electrical Machines and Transformers



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Influence of the Correlated Location of Cores of TPZ Class Protective Current Transformers on Their Transient State Parameters Elzbieta LESNIEWSKA and Wieslaw JALMUZNY Department of Applied Electrical Engineering & Instrument Transformers, Technical University of Lodz, ul. Stefanowskiego 18/22, 90-924 Lodz, Poland [email protected], [email protected] Abstract. A method of avoidance of mutual influences between secondary windings of multi-core type in a designed protective current transformer is presented in this paper. The analysis was performed for different location of cores with secondary windings. The mutual influence is determined on the basis of 3D field distributions obtained by the numerical field method. Some results of computation were compared with test results.

Introduction Protective current transformers are very important parts of electric power systems. They are indispensable for the proper functioning of a system, because they are elements of the protection system. There are two kind of protective current transformers; class P to protection at steady state and TP to protection at transient state. The multicore type current transformer is composed of a number of cores with individual secondary windings and a common primary bar in the same casing. During a transmission line short circuit, the primary current takes on an exponential component resulting in core saturation and a deformation of the secondary current. The cores of measuring current transformer and protective class P should be saturated during a transmission line short circuit. Therefore the measuring current transformer has a core without air gaps. The TPZ class protective current transformer has a core with air gaps which guarantee linearity of magnetic characteristic of the core at an assumed value of primary short circuit current Ipsc. Behaviour of protective current transformer at a transient state is very important because it influences the proper functioning of the protection system. The mutual influence between windings can occur through the magnetic field. The aim of research was to estimate the influence of the correlated location of cores on current error and the phase displacement at rated state and transformation errors at transient state. During the design process of the multi-core type current transformer it is important to predict the mutual coupling between the current transformers and then to avoid it.

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A Mathematical Model The field-and-circuit method for steady state, based on the solution of Helmholtz equation for 3D electromagnetic field determines secondary voltage and compares it with the circuit equation of the same secondary voltage. It is impossible to solve the transient problem using the complex method, since the wave-shapes of field quantities are considerably deformed. Joining field-and-circuit method and space-time 3D analysis allows computing the secondary current vs. time assuming a non-sinusoidal wave-shape of the primary current. For 3D analysis a full set of time-dependent differential equations must be solved instead. ⎛ ∂A ⎞ curl (ν curl A ) = σ ⎜ − − gradV ⎟ ⎝ ∂t ⎠

(1)

⎛ ⎛ ∂A ⎞⎞ div ⎜ σ ⎜ − − gradV ⎟ ⎟ = 0 ⎠⎠ ⎝ ⎝ ∂t

(2)

Using the commercial software 3D based on the numerical finite element method enables solving this equation and estimating the secondary current vs. time while the primary current has an exponential component. The non-linear magnetic characteristic of cores was taken into account. The boundary conditions were A × n = 0 and

V=0

(3)

at the boundary of the whole system with the surrounding air. The applied time stepping was 0.0001 s. The mesh of 616043 elements for 3D model was result of accuracy analysis. Further mesh refinement does not change the solution. Computational Results As an example, the three toroidal protective current transformers TPZ class 1200 A/1 A with eight air gaps of δ = 3 mm were considered. The total length of air gaps is 24 mm. The air gaps are rotated one by one at 45 degrees.

Figure 1. Multi-core type current transformer composed of three cores with individual secondary windings and a common primary bar.


a)

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Figure 2. Cores with gaps in the same position and the cores are turned at 22.5 degrees with relation to each other.

The neighbouring cores with secondary windings can be located in a different way. Two border cases were considered in work; one when the cores with gaps are in the same position and the second when the cores are turned at 22,5 degrees to each other. The analysis was performed for rated steady state and transient state. The current error and the phase displacement indicate the accuracy class of designed current transformer. The computations of current error and phase displacement were performed at rated state for load R = 5 Ω. The measurements were performed in the same conditions using the type Φ5304 measuring bridge with comparator produced by the company POCTOK. The test for the single TPZ class protective current transformers 1200 A/1 A gives the following results: the current error –0.92% and the phase displacement 398minutes. The results of 3D analysis carried out for the individual TPZ class protective current transformers 1200 A/1 A were: the current error –0.88% and the phase displacement 346 minutes. Table 1 shows results of computation for two different positions of cores. Both constructions give convergent results of errors. The middle protective current transformer has somewhat better conditions if the cores are turned at 22,5 degrees to each other. The computations of the instantaneous error current vs. time were performed at transient state for load R = 5 Ω, specified primary time constant Tp = 50 ms and rated symmetrical short-circuit current factor Kssc = 25. The primary current was equal ip = 30 2 (cos314.16t–e–20t) kA. In Fig. 5 the results of the computed instantaneous error current vs. time at a transient states are presented for three protective TPZ current transformers. All curves are very close. For both case the magnetic field linked between current transformers is practically negligible. In Fig. 4 we can observe that the both constructional solution give practically negligible magnetic flux density in the neighbouring core caused by the other protective current transformers.

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a)

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Figure 3. Distribution of magnetic flux density [T] of protective current transformer at rated steady state at the same conditions for two different positions of cores. Table 1. The current error and the phase displacement of the TPZ current transformer at rated state Rb = 5 Ω TPZ class protective current transformers 1200A/1A (δ = 24 mm) gaps in the same position core: current error ΔI [%] phase displacement δi [min]

outside –0.887 342

middle –0.883 347

outside –0.880 342

gaps turned at 22.5° outside middle outside –0.828 –0.800 –0.825 341 334 341

Peak instantaneous alternating current errors have been determined on the basis of obtained curves (Fig. 5)

εâc =

iˆε ac ×100% 2 I psc

(4)

where Ipsc = KsscIpn = 25 * 1200 = 30 kA, îεac – maximum instantaneous error of the alternating current component and is equal 9.7%, and have been determined on the basis of test is equal 11.4%. Test was carried out using the d.c. saturation method.

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Figure 4. Distribution of magnetic flux density [T] of protective current transformers at transient state for two different positions of cores at the same conditions and time t = 0.054 s.

25 20 15 10 5 0 -5

0

0,05

0,1

0,15

0,2

time (s)

outside

middle

outside

Figure 5. Instantaneous error current vs. time for three protective current transformers (in terms of secondary winding).

236


a)

b)

Figure 6. Distribution of magnetic flux density [T] of the two P class protective current transformers and one TPZ class protective current transformer in the middle at rated steady state a) three current transformers together b) only the TPZ class protective.

The next problem was recognizing the influence of the two neighbouring protective current transformers type 10P 45 1200 A/5 A on operation of TPZ clas protective current transformer 1200 A/1 A. The same computation was performed, for the case with different class of current transformer. The computations of current error and phase displacement were performed at rated state of the TPZ class protective current transformer for load R = 5 Ω and the 10P class protective current transformers of the accuracy limit factor ALF = 45 for load S = 15 kV and cosφ = 0.8. In Fig. 6 can be observe that the cores of both P class protective current transformers are more saturated but the TPZ class protective current transformer with core with air gaps works in normal condition. Its current error and the phase displacement practically did not change.

237


Table 2. The current error and the phase displacement of different class of current transformers at rated state class of current transformer:

10P 1200 A/5 A

current error ΔI [%] phase displacement δi [min]

–0.099 +7,70

TPZ 1200 A/1 A ° –0.836 +343

10P 1200 A/5 A –0.099 +7,70

a)

b)

Figure 7. Distribution of magnetic flux density [T] of the two P class protective current transformers and one protective current transformer TPZ class in the middle at transient state at the same conditions and time t = 0.054s a) three current transformers together b) only TPZ class protective.

Research shows significant difference between protective current transformers class P and TP (Fig. 7). Nevertheless that the accuracy limit factor of them is very big (45) and symmetrical short circuit current factor Kssc during test is 25, P class protective current transformer cores are saturated and the current transformation are incorrect, in opposite to TPZ class protective current transformer (Figs 8, 9). It causes by an exponential component of the primary current.

238


secondary current (A)

130 80 30 -20

0

0,02

0,04

0,06

0,08

0,1

-70 -120 -170 -220 time (s) TPZ

P

P

Figure 8. Secondary currents vs. time of TPZ and P class protective current transformers.

instantaneous error current (A)

25 20 15 10 5 0 0

0,02

0,04

0,06

0,08

0,1

-5 time (s)

instantaneous error current (A)

sigle TPZ

TPZ nearby P

350 300 250 200 150 100 50 0 -50 0

0,02

0,04

0,06

0,08

0,1

time (s) Type P

Figure 9. Instantaneous error current vs. time for two type of protective current transformers a) a comparison curves for separated TPZ class current transformers and working in neighbourhood of P class protective current transformers b) for P class protective current transformers.

Conclusions The transient state 3D analysis of protective current transformers performed with the application of the field-and-circuit method can determine the correlated location of cores, which guarantees a magnetically separated operation of each core with individual secondary winding. The computing results show if correlation location of cores has an influence on steady state and transient errors. To eliminate the mutual interactions during the design


239

process, the construction may be redesigned if the steady state and transient state errors are excessive. The tests show that in these cases the TPZ class protective current transformers can operate as independent devices. This means that the neighbouring current transformers have only a slight influence on their errors.

References [1] E. Lesniewska, Applications of the Field Analysis During Design Process of Instrument Transformers, Transformers in Practice, Vigo Spain 2006, pp. 227-251. [2] E. Lesniewska and J. Ziemnicki, Transient State Analysis of Protective Current Transformers at Different Forced Primary Currents, Przegląd Elektrotechniczny 5’2006, pp. 57-60. [3] W. Jalmuzny, D. Adamczewska, I. Borowska-Banas, Analysis of Current Difference Test Arrangement Operation for Measuring Class TP Current Transformers, Pomiary Automatyka Kontrola (PAK), no 10bis/2006, pp. 102-110. [4] E. Lesniewska and W. Jalmuzny, Influence of the Number of Core Air Gaps on Transient State Parameters of TPZ Class Protective Current Transformers, Compumag 2007, Aachen. This work was supported by the Polish Ministry of Science and Higher Education (Project No. 3 T10A 004 30).

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Machine with a Rotor Structure Supported Only by Buried Magnets Jere KOLEHMAINEN ABB Oy, Motors, P.O. Box 633, FI-65101 Vaasa, Finland E-mail: [email protected] Abstract. A buried magnet rotor structure, which is supported only by permanent magnets, is proposed for medium speed permanent magnet machines. A machine utilizing the construction is built, tested and compared to another machine with traditional V-shaped poles. The machine is also simulated using Finite Element Method and the results are compared to tested values. The obtained results demonstrate the feasibility of the construction.

Introduction Permanent magnet synchronous machines (PMSM) with buried magnets have been considered in a wide range of variable speed drives. A buried magnet design has many advantages compared to designs with surface mounted and inset magnets. With a buried magnet design flux concentration can be achieved, which induces higher air gap flux density [1,2]. That, in turn, gives a possibility to increase torque of a machine. The typical way of manufacture a buried PM rotor is to assemble a stack of punched rotor disks with rectangular holes and insert magnets into these holes. The rotor poles between the magnets are fixed to rest of the rotor structure with thin iron bridges. The disadvantage of the supporting bridges is the leakage flux, the magnitude of which depends on the thickness of the bridges. In low speed applications this is not a problem, since the centrifugal forces acting on the poles are relatively small and the bridges can be kept thin. However, as the tangential speed of the rotor surface in medium speed applications (4000…8000 1/min) exceeds 60 m/s (corresponding to 4000 1/min in machine size IEC250) the stress in the bridges will exceed the yield strength of the electrical steel (typically 300 MPa for grade M400-50A). The problem can be countered by increasing the thickness of the bridges, however, this increases the leakage flux, which in turn increases the amount of magnet material needed to get the required torque. However, there exists a solution with thinner bridges, where magnets are partly used to support the pole structure [3]. In this paper we go further and study a solution on how to get mechanically more robust rotor structures without using iron bridges. In the solution the tensile stress is geometrically converted into compressive one and only the magnets are used to support the pole structure. The new solution is compared to a traditionally used solution with V-shaped poles. The comparison is done using time stepping and static calculations using Finite Element Method (FEM) [4]. Machines with both the rotor designs are built and tested. The machine with the new dovetail pole design is analyzed further and results are compared to simulations.

J. Kolehmainen / Machine with a Rotor Structure Supported Only by Buried Magnets

241

Figure 1. Designs with dovetail and V shaped poles with flux lines created by the flux of the magnets.

Machine Designs An 8-pole machine with V-shaped rotor poles is used as an example for comparison to a machine with the new dovetail design without supporting bridges. The machine has shaft height 250 mm, nominal power 110 kW, voltage 370 V, and speed 4800 1/min. The only difference of the two machines is in their rotor structure as it can be seen in Fig. 1. An 8-pole machine has 8 symmetry sections in V-pole design, but in the dovetail design the rotor has magnets in every second pole [1]. With both designs, volume of the magnets and dimensions of the magnets seen by stator are same. With the V shape design, total magnet width and length in one pole are 2 × 7.3 = 14.6 mm and 52 + 52 = 104 mm and with the dovetail design, these are 1 × 14.6 = 14.6 mm and 26 + 52 + 26 = 104 mm. Length of the both rotors is 120 mm. Magnetically, there are two major differences which affect to electrical properties of machines. The dovetail design has not magnetic bridges between poles so the leakage flux is reduced especially with low saturation of flux. Every second pole of the dovetail design has a different tangential air gap length so electrical properties with high load angles are expected to be slightly worse than with the V shape design.

Manufacturing Two machines with both rotor types are manufactured. The general method to manufacture the rotor with a V shaped design is to assemble a stack of disks, compress it using bolts and nonmagnetic end plates and shrink fit the stack on the shaft. Then, the magnets are inserted into their holes using glue. The rotor with the dovetail design is manufactured with a different method, which is to assemble five (one central body and four small poles) stacks of disks, compress them using bolts and nonmagnetic end plates and shrink fit the central body stack on the shaft. Next, the magnets are fixed to four pole stacks using glue. Resulting poles are axially inserted to central body stack using glue.

242


Figure 2. Manufacturing rotor with dovetail design. In the left there is a pole with magnets, in the middle there is the rotor without the pole, and in the right the pole is inserted to rotor. Table 1. Open Circuit Voltages Quantity

Dovetail design

V-shape design

311.1 365.5

330.5 347.6

Measured voltage (V) Calculated voltage (V)

Results The machines with the both designs are tested and analyzed. For all load tests, as for our industrial cases, the direct torque control strategy with software for permanent magnet AC machines is used with frequency converter ACS600 [5]. All electromagnetic calculations are done with time stepping Finite Element Method [4]. In load calculations, voltage source is used, the form of the voltage is sinusoidal and amplitude is kept the same. Simulations are started with various rotor angles without initial solution and stopped after 41 electric periods when transient oscillations have totally died away. Constant rotor speed is used. Iron losses are calculated from the equation 2

PTOT = kh Bm2 f +

1 T ⎡ d 2 ⎛ dB ⎞ dB ( t ) ⎟ + ke ⎛⎜ ( t ) ⎞⎟ ⎢σ ⎜ ∫ 0 T ⎢⎣ 12 ⎝ dt ⎠ ⎝ dt ⎠

32

⎤ ⎥ k f dt , ⎥⎦

(1)

where Bm is the maximum flux density at the node concerned, f is the frequency, σ is the conductivity, d is the lamination thickness, k h is the coefficient of hysteresis loss and k e is the coefficient of excess loss. Open Circuit Voltage Measured open circuit voltages of both machines, at speed 4800 1/min, are compared to the calculated ones in the Table 1. The measuring and the calculating temperature has been 20 oC. In this case, the magnets have a remanence flux density of 1.1 T and energy product 230 kJ/m3. The dovetail design has 5.2% larger calculated open circuit voltage than the V shape design. The measured open circuit voltage is 4.9% smaller than calculated for the V shape design and it is 14.9% smaller for the dovetail design. The measured open circuit voltages were expected to be smaller, because small rotor length and diameter


243

Figure 3. Open voltage harmonics and torque ripples with speed 4800 1/min of dovetail and V shape designs.

Figure 4. Calculated torque and current (left) and efficiency and power factor (right) as a function of electric load angles with dovetail and V shape design.

ratio (120/289 = 0.42) causes remarkable leakage fluxes in ends of rotors. With the dovetail design, also axial deviations of the rotor will add leakage fluxes (the rotor body is slightly longer than the poles). The machine with the dovetail design has also different harmonic distribution of open circuit voltage, as can be seen in Fig. 3. The fifth harmonic is almost same with both designs. The seventh, eleventh and thirteenth harmonics are larger with the dovetail design while the fifth, seventeenth and nineteenth harmonics are larger with the V shape design. This is caused by different air gap forms and different sizes of every second pole with dovetail design. Practically, voltage and torque ripples are on the same level. Calculated Electrical Properties with Different Loads The calculated electrical properties as a function of electric load angle are compared in Fig. 4. With the dovetail design the torque is sinusoidal; the reluctance torque is negligible and the maximum torque 382 Nm with load angle 90 degrees is smaller than torque (414 Nm) with the V shape design, because of asymmetric pole pairs. With the V shape design, maximum torque is 426 Nm at load angle 102 degrees. Furthermore, maximum reluctance torque is 49 Nm. In addition, the current behaves differently, because of different saturation. With the dovetail design, power factor is larger with electric load angles under 55 degrees and with higher load angles, it is slightly smaller. However, efficiencies are slightly better with the V shape design.

244


Figure 5. Calculated and measured input and output power with dovetail (left) and V shape (right) designs as a function of current. Table 2. Comparison of Nominal Load Results Quantity Shaft Power (kW) Torque (Nm) Voltage (V) Current (A) Efficiency Power Factor Total Losses (kW) Copper Losses (kW) Total – Copper (kW) Iron Losses (kW) Other Losses (kW)

Measured Dovetail design

Calculated Dovetail design

Measured V-shape design

Calculated V-shape design

110.2 219.1 370 229.7 0.947 0.791 6.26 1.66 4.60

110.4 219.6 370 194.0 0.950 0.935 6.26 1.19 5.08 1.50 3.58

111.1 221.0 370 212.9 0.950 0.858 5.90 1.30 4.60

110.3 219.5 370 202.8 0.946 0.898 6.03 1.18 4.85 1.27 3.58

Comparisons with Different Loads Measured input and output powers as a function of current are compared to calculated ones in Fig. 5. Generally, measured powers are smaller than calculated powers. The possible reason is the same than with the case of voltages; leakage fluxes in the ends of rotors. Difference is also bigger with the dovetail design. Same differences can be seen in comparison of nominal load results. Nominal Load The measured and the calculated nominal load results of the two designs are compared in the Table 2. Used stator winding temperature is 75 oC. Approximated “Other Losses” contains all other losses except friction and additional losses. Iron losses are calculated with Eq. (1). All efficiencies have the same magnitude. With the dovetail design, the calculated power factor is 4.1% better and measured power factor is 7.8% worse than with the V shape design. Torque oscillations are compared with different electric load angles in Fig. 6. The oscillations are larger with the dovetail design and calculated load angles over 8 degrees. Oscillation is studied further with load angle 40 degrees in Fig. 6. Clear sixth order torque harmonics can be seen with the dovetail design. With the V shape design, remarkable twelfth order torque harmonic reduces total oscillation.


245

Figure 6. Calculated torque oscillations as a function of electric angle (left) and torque as a function of time with electric load angle 40 degrees (right) with dovetail and V shape designs.

Figure 7. Von Mises stress with speed 4800 1/min. The stresses are greatest in dark grey areas.

Strength of Structures Stress Analysis The rotor with the new dovetail design has a totally different stress distribution compared to the V-pole rotor. In the V-pole rotor, all of the shear and tension stresses are in the iron bridges whereas in the dovetail design, most of stresses are compression stress in the magnets and shear stress near corners of magnets. Von Mises stresses in the dovetail and in the V shape designs with are shown in Fig. 7. Computation is done using the centrifugal force associated with the speed of 4800 1/min. The largest stress in electric steels of the dovetail design, 130 MPa, is locally in the corners of sheets. With the V shape design, average stress in the inner bridges is 90 MPa and the largest stress in electric steels, 200 MPa, is also locally in the corners of sheets. These values are below the yield strength (305 MPa) of the steel. With the dovetail design, in center of the smaller magnets the stress is 50 MPa. It is well below the maximum compressive strength of the magnets. The calculated maximum stress in magnets is 381 MPa (located in corners). Using magnets to compose the structure it becomes robust enough for the speed of 4800 1/min.

246


Stability of Rotor with Dovetail Design The mechanical durability test of the motors consists of two load runs with speed 4800 1/min. First, the motor is driven with different loads and operating temperatures for one and half hours. After this, the motor was cooled over night. Finally, temperature test has done for four and half hours. Measured vibration levels remained same thought all tests. Visual check was done after test. The magnets remained solid. The glue seam between inner magnets and rotor body was separated. In the sides of smaller magnets, glue was changed its color from grey to light grey. This indicate that glue was deformed, not separated. Hence, stability of the rotor remained, but more tests should be done to see whether stability remains with longer period.

Conclusion The prototype machine with a dovetail-shaped magnet poles exhibits a significant increase in mechanical stability over the conventional V-pole design. By converting the tensile stress in the iron bridges into a compressive stress in the magnets by redesigning the pole geometry, a very robust construction can be achieved. The electrical properties and the consumption of magnetic material can be kept on the same level as in the V-pole design.

References [1] Kolehmainen J., “Finite Element Analysis of Two PM Motors with Buried Magnets”, ICEM, Krakow, Poland, 2004, SPRINGER MONOGRAPH “Recent Developments of Electrical Drives”, Nov. 2006. [2] Ohnishi T., Takahashi N., “Optimal design of efficient IPM motor using finite element method”, IEEE Trans. Magn., vol. 36, no. 5, 3537-3539, Sep. 2000. [3] Kolehmainen J., Ikäheimo J., “Motors with Buried Magnets for Medium Speed Applications”, IEEE Trans. Energy Convers., to be published. [4] Flux2D software – www.cedrat.com. [5] ACS 607-0400-5, frequency converter – www.abb.com.


247

FEM Study of the Rotor Slot Design Influences on the Induction Machine Characteristics Joya KAPPATOU, Kostas GYFTAKIS and Athanasios SAFACAS University of Patras, Department of Electrical and Computer Engineering, Electromechanical Energy Conversion Laboratory, 26500 Rion Patras, Greece E-mail: [email protected] Abstract. A library of parameterized Squirrel Cage Induction Machine models has been constructed regarding the rotor slot design and is used to investigate the influences of the rotor bars shape on the machine characteristics. The slot shapes chosen correspond to standard induction machines used for specific applications. The calculations were conducted using the Finite Element Model (FEM) of the machine and the torque and current waveforms against speed, as well as the field distribution and the copper losses in the rotor bars for every model at starting have been obtained. Useful conclusions about the influences of the slot design on the Induction Machine behavior, mainly at starting are derived.

Introduction As known, the rotor of Induction Machines has stronger effect than the stator on the performance characteristics of the machine and especially on its starting performance. On the other hand, there are specific limits of the machine variables, e.g. maximum value for the starting current and minimum values for the starting torque, pull-up torque and breakdown torque, depending upon its ratings. Thus, different design classes for cage Induction Motors are set, each one suitable for different performance requirements and specific applications. These design classes refer to the rotor design and specially to the geometrical characteristics of the rotor slots. Furthermore the interest of both the designers and the users is directed to the optimization of the machine performance and the increase of the efficiency. For that reason several papers have been published during the last decades on the analysis and optimization of cage Induction Motors and specially on the suitable design of the rotor slots, which is critical for the starting performance and the shape of the torque-speed curve [1–5]. In this paper in order to compare more accurately the performance of cage Induction Machines of different rotor slot design, the 2-d Finite Element Method is used. Four different rotor slot designs are used, which are presented schematically in Fig. 1 and they refer to standard squirrel cage motors used for specific applications. The geometrical variables of the rotor cross section have been parameterized and a FEM code, which is used to model the motor has been developed. In order to obtain a more flexible and user friendly code, a library of four parameterized models has been constructed, which differ between them as for the shape of the rotor slots. As the shape of the slot remains the same in each model, the slot dimensions can easily and accurately be modi-

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J. Kappatou et al. / FEM Study of the Rotor Slot Design Influences

a)

b)

c)

d)

Figure 1. Characteristic rotor slots designs.

fied. In all cases the stator model, the air-gap length, the stator voltage and frequency, as well as the number of rotor slots are the same in order to compare the influences of the rotor slots design on the operational characteristics of the machine. Besides, as the rotor core is usually slightly saturated and also the skin effect plays an important role, specially in the deep-bar design, the same B-H curve of the ferromagnetic material and a suitable number of Finite Elements in the slot area are used. In all these cases the distribution of the field in the core and the waveforms of the torque and current as functions of the speed are calculated and useful results of the rotor slots shape effect on the machine behavior are derived. The use of the models library results in a flexible and more friendly code, which can be used for the application of optimization techniques for every specific design type, as fewer design geometrical variables are used than in a model of arbitrary slot shape.

FEM Analysis The numerical analysis is based on a magnetic vector potential formulation and a commercial package, OPERA of VF, has been used for the Finite Elements Analysis. The model can include non-linear materials as the magnetic saturation plays an important role, mainly in the rotor of the machine. In order to model the skin effect, specially in the cases of deep bars and double cage a suitable mesh has been constructed in the bars area. In the above cases, high currents are induced in the rotor bars, specially at starting and the skin effect must be taken into account. The stator windings are fed from sinusoidal voltage sources, which are connected via external circuits to the model. A library of four parameterized models of Asynchronous machines has been constructed, which differ between them as for the shape and the dimensions of the cage bars. Each design from now on will be reported with the letters a,…,d according to Fig. 1. All models have the same stator with 36 slots and 2 pole pairs, the same value and frequency of the supply voltage and 48 rotor slots. Although the dimensions of the rotor bars were selected arbitrarily for the various models, some parameters which affect the motor performance are computed and some useful qualitative conclusions regarding the influence of the bars design on the motors behaviour can be derived. These are calculated mostly for the starting, which is critical for the machine performance.

Simulation Results As known the design of the rotor bars affects strongly the motor performance and specially its starting. An arbitrary bar design can result to higher torque and less current


a)

b)

c)

d)

249

Figure 2. Flux lines and current density in the bars at starting for the various models of the slots according to Fig. 1: a) bar model a, b) bar model b, c) bar model c, d) bar model d.

Figure 3. Flux lines and current density in the bars near synchronous speed for the model of the double cage, bar model d.

during starting, but to much greater slip and consequently to a reduction of the efficiency at nominal load. In this work the dimensions of the rotor bars were selected arbitrarily for the various models and we can not achieve general conclusions. However some parameters which affect the motor performance, mainly at starting, are computed and some useful qualitative conclusions regarding the advantages and disadvantages of each model can be derived. In Fig. 2 the flux lines at starting on a cross section of the machine for the various models of the slots, according to Fig. 1, are presented. In the same plot the current density in the bars is superimposed also. As the bar becomes deeper, Figs a and d, the leakage of the rotor increases and less flux is penetrating into the rotor body. At starting the skin effect plays an important role because the skin depth is at its smallest value. The distribution of the current density in the bars, specially in the case of the double cage, Fig. 2d, shows clearly the skin-effect. In Fig. 3 the flux lines and

250


β

γ

Figure 4. Spatial distribution of the amplitude of current density along the axis (j-k) of a rotor bar, in A/mm2, at starting for the different bars used: α) the type of bars a and b, β) the bar type c for two values of the conductivity of the bars, γ) the bar type d for the linear and the non-linear B-H curve.

the distribution of the current density in the bars is presented for the case of the double cage near the synchronous speed (1 Hz) for comparison with the starting, Fig. 2d. For a better display of the skin effect, the spatial distribution of the current density along the axis j-k of a rotor bar at starting is demonstrated in Fig. 4. In all models the same bar has been selected for the calculations. The value of the electrical conductivity of the rotor bars has been taken equal to 60 kS/ mm, except for the bar design c, where a second case of less conductivity, 40 kS/mm, has been analyzed also for comparison. The current density increases in the top of the slot, specially for the case of double cage, Fig. 4γ, as a result of the skin-effect. In Fig. 4β the influence of the electrical conductivity of the bars on the current density in the bar type c is demonstrated. When a non-linear B-H curve is used in the model of the motor with double cage, bar type d, the current density is gradually increasing along the axis of the bar comparing to the linear case, essentially in the top of the slot near the air-gap. The effect of the bar design on the waveforms of the torque and stator current against speed is presented in Fig. 5. The motor design c, which develops the highest starting torque is a high slip motor, while the smallest value of starting current appears in the motor design d. Although the dimensions of the rotor bars were selected arbitrarily for the various models, some more motors parameters which affect the motor performance are computed and some useful qualitative conclusions regarding the advantages and disadvantages of each model can be derived. These are calculated at starting and are presented in the following Table 1. From Fig. 5 and Table 1 the following useful conclusions are obtained: •

The starting current is lower in the case of the double cage, type d, comparing with the motor of bar type c. This can be explained from the higher value of

251


35

c

d

a

b

c

d

50

a

b

30 40

30

20

I(A)

M (Nm)

25

15

20

10 10

5 0

0 0

200

400

600

800

1000

1200

1400

1600

0

200

400

600

800

1000

1200

1400

1600

n (rpm)

n (rpm)

a

b

Figure 5. Waveforms for the various designs of the bars of: a) the torque, b) the stator current. Table 1. Performance characteristics for the different rotor slots design at starting

0.268 0.282 0.679 1.012 0.853

Amplitude of current in the stator (A) 45.352 48.4 35.63 29.8354 19.86

Total ohmic losses in the bars (W)/1bar resistance (mΩ) 5856.302/0.0938 6812.75/0.09609 8907.686/0.2354 9202.5/0.3527 4526.206/0.425

Mean ohmic losses per surface in the bars (W/mm2) 2.8632 3.7955 14.944 15.4385 3.372

1.079

22.5

5863.27/0.425

4.368

Bar type according to Fig. 1

Torque (Nm)

Torque/Is (Nm/A)

a b c, conduc. = 60 kS/mm c, conduc. = 40 kS/mm d, linear B-H

12.18 13.68 24.2 30.212 16.957

d, non-linear

24.29

•

•

the rotor leakage and consequently the leakage reactance in the first case, as observed in Fig. 2 c and d. Although the starting torque is greater in case c, the ratio of starting torque per current presents higher value in the bar type d, particularly when the magnetic saturation is taking into account in the model, the above ratio increases more. The decrease of the conductivity of the bars in the bar design c, which results in higher bar resistance, increases the starting torque and limits the starting current. Particularly the ratio of the starting torque per ampere increases from 0.679 to 1.012. The mean ohmic losses per surface in the bars in the design c are much greater than in any other case. In Table 1 also appears the resistance of a cage bar, as calculated, for every model.

Conclusions A library of four parameterized Squirrel Cage Induction Motors models of different bars shapes has been constructed and an investigation of the effect of the bars design on the characteristics of the motor has been carried out. The waveforms of torque and current as function of speed are calculated and compared for all models. Moreover the distribution of the field and some more parameters at starting, like the variation of the

252


current density along the axis of a bar, as well as the ohmic losses in the cage have been computed, which give us some useful qualitative conclusions regarding the advantages and disadvantages of each model.

References [1] M. Nurdin, M. Poloujadoff, A. Faure, “Synthesis of squirrel cage motors: A key to optimization”, IEEE Trans. on Energy Conversion, Vol. 6, No. 2, pp. 327-335, June 1991. [2] S. Williamson, C. McClay, “Optimization of the geometry of closed rotor slots for cage Induction motors”, IEEE Trans on Industry Applications, Vol. 32, No. 3, pp. 560-568, May/June 1996. [3] M.R. Feyzi, H.V. Kalankesh, “Optimization of Induction motor design by using the finite element method”, CCECE/CCGEI 2001. [4] C. Grabner, “Investigation of Squirrel cage Induction motors with semi-closed and closed stator slots by a transient electromechanical finite element technique”, ISEF 2005, Baiona, Spain, conference proceedings record, September 2005. [5] Min-Kyu Kim, Cheol-Gyun Lee, Hyun-Kyo Jung, “Multiobjective optimal design of three-phase Induction Motor using improved evolution strategy”, IEEE Trans on Magnetics, Vol. 34, No. 5, pp. 2980-2983, September 1998.


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Concentrated Wound Permanent Magnet Motors with Different Pole Pair Numbers Pia SALMINEN, Hanne JUSSILA, Markku NIEMELÄ and Juha PYRHÖNEN Lappeenranta University of Technology, Department of Electrical Engineering, P.O. Box 20, 53851 Lappeenranta, Finland [email protected] Abstract. The study addresses the torque production capabilities and losses of concentrated wound permanent magnet machines. Different permanent magnet synchronous motors are modelled with both semi-closed slots and open slots. With open slots, the coils can be easily assembled to the stator, thus making it attractive to study the performance or benefits of the concentrated wound PM machines with open slots. Different slot and pole combinations are considered on the basis of the finite element analysis (the Flux2D program package by Cedrat being employed in the computations).

Introduction This work examines the performance of different model designs for a machine with a rated torque of 1075 Nm, a frame size of 225, and a rated speed of 400 rpm. These values represent ratings for machines used for example in the paper making industry. To compare the machines, the electromagnetic losses are calculated to estimate the efficiency of each machine. Particular attention is paid to the iron losses, Joule losses (I2R losses), and eddy current losses caused by the permanent magnets of concentrated wound PM machines. Concentrated wound machines are one type of fractional slot wound machines. The number of slots per pole and per phase q ≤ 0.5. In a concentrated wound machine, each coil is wound around one tooth to achieve as short end windings as possible. This reduces the amount of copper and leads to low Joule losses by virtue of the short end winding. It is also possible to utilize short end windings by inserting a longer stator stack compared to the stack of integer slot wound machines in the same frame size. Consequently, longer active parts will give more torque. Figure 1 illustrates a concentrated wound prototype machine, constructed within a size 225 frame. In this case, the end windings are so short that it is possible to make the stator stack 30% longer compared to the stack length of a four pole integer slot wound induction motor [1–3].

Finite Element Analysis A set of finite element analyses (FEA) is performed to estimate the pull-out torque and the losses of each design. The pull-out torque is the maximum torque that a motor can sustain at synchronous operation. The finite element computations are based on Cedrat Flux2D [4] program package using transient analysis for model concentrated wound

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P. Salminen et al. / Concentrated Wound Permanent Magnet Motors

end windings

270 mm

end windings

D

Stator stack

Figure 1. Concentrated wound PM machine in a size 225 frame. The short end windings leave more space inside the frame; hence, the stator stack may be designed longer in the axial direction.

a)

b)

Figure 2. Concentrated wound PMSM a) with a low pole pair number and b) with a high pole pair number.

machines with different pole pair numbers, p. Figure 2 (a) shows a machine with a low pole pair number p = 4, and 12 slots. For this machine, the number of slots per pole and per phase q = 0.5. In Fig. 2 (b), the pole pair number is high (p = 15), there are 36 slots, and q = 0.5. It has been noticed in previous studies [5] that with q = 0.5 the torque production capability is higher than with q < 0.5. When q = 0.5, the cogging torque and torque ripple may be higher than with other possible concentrated wound combinations, while it is still possible to minimize torque ripples by optimising the magnet width [6]. Semi-Closed Slots The machines under investigation have surface mounted magnets as shown in Fig. 2. Most of the models are constructed with semi-closed slots in the stator, except for certain special cases that have open slots. The boundary conditions for machine dimensions are determined by the size 225 frame; the outer diameter 364 mm and the machine length 270 mm. The rated speed is 400 rpm and the rated torque is 1075 Nm. The total losses of concentrated wound PM machines calculated are presented in Fig. 3. The total losses comprise the iron losses, Joule (I2R) losses, and eddy current losses caused by the permanent magnets. It can be seen in Fig. 3 that the Joule losses dominate by a large margin, but their proportion of the total losses decreases when p is large. Consequently, if low Joule losses are an important design parameter, it is advisable to avoid low pole pair numbers. However, the iron losses increase as the number of poles increase, which is also expected as the frequency increases. The proportion of


255

Figure 3. Iron losses, Joule losses, and eddy current losses caused by permanent magnets for machines with surface mounted magnets having semi-closed slots. The results are obtained by the FEA.

iron losses to the machines in this study rises from 0.3 kW to 1 kW as the frequencies increase from 27 to 140 Hz. Please note that the minimum net losses occur when p = 10 (see Fig. 3). The machine with p = 10 has about 25% lower net losses than p = 4 and p = 21 machines. The model parameters for the magnet material are based on the values for Neorem 495a100, the remanence flux density of which is Br = 1.05 T and the coercivity HC = 800 kA m–1. The resistivity of the sintered magnets permits remarkably large eddy current losses if the flux density may pulsate in the magnet during the motor running. This phenomenon makes concentrated wound machines quite vulnerable to large eddy current losses in the magnets. In integer slot wound machines with semi-closed slots the flux density pulsation in the magnets may be negligible compared to fractional slot machines with concentrated windings. The aim here is to investigate what kind of a role the permanent magnet eddy current losses have at low speed (e.g. 400 rpm). Usually, a machine with geometrically wide magnets generates high eddy current losses, while small or narrow magnets generate lower eddy current losses [7]. Figure 4 shows the eddy current losses caused by permanent magnets for several surface-mounted PM machines. As expected, the machines with low pole pair numbers have high eddy current losses because of their large magnets in the rotor. Geometrically large surfaces are harmful with respect to the eddy current losses. Semi-Closed Slots Compared with Open Slots Next, the effect of the slot opening width on the losses of PM machines is examined. The above semi-closed machines are now modified so that they have open slots. The results obtained by the FEA are shown in Fig. 5. The results demonstrate that only the iron losses are smaller with the open slot structures than with the semi-closed slot structures. The Joule losses turn out to be higher with open slots than with semi-closed slots, because more winding turns are needed in the stator slots to induce sufficient back electromotive force in the system. With open slots, the eddy current losses caused by permanent magnets are approximately double compared with those of motors using

256


500 12 Slots 24 Slots 36 Slots

Losses (W)

400 300 200 100 0 0

4

8

12

16

Pole pair number, p Figure 4. Eddy current losses caused by permanent magnets for machines with surface-mounted magnets having semi-closed slots. Results are obtained by the FEA.

Figure 5. Iron losses, Joule losses and eddy current losses caused by the permanent magnets for machines with surface mounted magnets having semi-closed slots and open slots. Results are obtained by the FEA.

semi-closed slots. For a machine with a low pole pair number (e.g. p = 5), the eddy current losses with semi-closed stator slots are calculated to be 400 W, while with open slots, the losses rise to 950 W. This result suggests that the eddy current losses in the case of concentrated wound PM machines should be evaluated carefully even at low speeds. With high pole pair numbers, the magnet dimensions are small, and therefore the eddy current losses are low. The calculated pull-out (maximum synchronous operating) torques for four pairs of model motor designs are shown in Fig. 6. Each of the designs has either open or semi-closed stator slots. For both types, when p = 21, the available torque is relatively


257

3

Pull-out torque (p.u.)

Torque

2

1

0 Semi

Open

12-slot-10-pole

Semi

Open

24-slot-16-pole

Semi

Open

36-slot-24-pole

Semi

Open

36-slot-42-pole

Figure 6. Pull-out torque for machines with surface mounted magnets having semi-closed slots and open slots. Results are obtained by the FEA.

small and not enough for appropriate motor operation. The machines with 16 poles and 24 poles have a high torque, and also their efficiency shows good performance. Open slot motors achieve slightly higher torque levels than the ones with semi-closed slots, except when p = 10; however, the efficiency with open slots is lower than with semiclosed slots.

Conclusion By applying finite element analytical methods, this paper examines several factors that affect the performance and efficiency of permanent magnet electric motors. The authors have, however earlier experimental data (e.g. [2]) that confirms the validity of the calculation process. Joule heating accounts for most of the losses; however, also eddy current effects play a significant role and must therefore be included in the analysis. For machines with rotor surface magnets, the electromagnetic energy losses are higher in the machines with an open-slot stator than in the machines with a semi-closed slot structure, especially when the pole pair number is low. This is a consequence of the flux pulsation in magnets and the large magnet surface area. It is possible to achieve a high pull-out (maximum) torque with both open slots and semi-closed slots by using an intermediate number of poles and slots; these designs also exhibit the lowest Joule losses. Since Joule losses are the dominant factor in the machines (machine design), the further studies will focus on embedded magnet motors. It is hoped that in these motors, the Joule losses will be low even when the number of pole pairs is large.

References [1] Cros J., Viarouge, P., Carlson, R. and Dokonal, L.V. 2004. Comparison of brushless DC motors with concentrated windings and segmented stator. Proceedings of the International Conference on Electrical Machines. ICEM 2004, Krakow, Poland. CD-ROM.

258


[2] Salminen, P. 2004. Fractional slot permanent magnet synchronous motor for low speed applications. Dissertation, Acta Universitatis Lappeenrantaensis 198, Lappeenranta University of Technology. 151 p. [3] Libert, F. and Soulard, J. 2004. Investigation on Pole-Combinations for Permanent-Magnet Machines with Concentrated Windings, Proceedings of the International Conference on Electrical Machines, ICEM 2004, Krakow, Poland. [4] Cedrat 2007. Software solutions: Flux®. [Online] Available from http://www.cedrat.com/ [Date accessed 26.6.2007]. [5] Salminen, P., Jokinen, T. and Pyrhönen, J. 2005. The Pull-Out Torque of Fractional-slot PM-Motors with Concentrated Winding, Electric Power Applications, IEE Proceedings, Vol. 152, Iss. 6, pp. 1440–1444. [6] Salminen P., Pyrhönen J., Libert F., and Soulard J. 2005. Torque Ripple of Permanent Magnet Machines with Concentrated Windings. 15–17 September 2005, International Symposium on Electromagnetic Fields in Mechatronics, ISEF 2005, Electrical and Electronic Engineering, Baiona, Spain. [7] Zhu, Z.Q., Ng, K., Schofield, N. and Howe, D. 2004. Improved analytical modelling of rotor eddy current loss in brushless machines equipped with surface-mounted permanent magnets. Electric Power Applications, IEE Proceedings. Vol. 151, Issue 6, 7 Nov. 2004 Page(s):641–650.


259

Air-Gap Magnetic Field of the Unsaturated Slotted Electric Machines Ioan-Adrian VIOREL, Larisa STRETE, Vasile IANCU and Cosmina NICULA Electrical Machines Department, Technical University of Cluj-Napoca, Daicoviciu 15, 400020, Romania [email protected], [email protected] Abstract. This paper deals with the air-gap magnetic field analytical calculation in the case of unsaturated slotted electric machines. Different analytical estimations of the flux density variation versus circumferential coordinate in the machine airgap are considered. The obtained results are compared between them and with the two dimension finite element method (2D-FEM) calculated values, where the iron core material nonlinearity is fully considered.

1. Introduction The air-gap flux density provides valuable information in evaluating slotted electric machine performance. Any method to design an electric machine, regardless its type, requires knowledge on the air-gap magnetic flux density to calculate the main dimensions, the necessary mmf, the torque (average, maximum, starting, cogging or ripple), the back emf value and shape and the main inductances, which may be dependent on the rotor position. It is clear by now that the common, and the most accurate way of obtaining the air-gap flux density is based on finite element method (FEM) calculation, but this is still time consuming even on powerful computers and it is difficult to use FEM in iterative design optimizing procedure, or to implement FEM results directly in on line control systems. The electric machines’ air-gap flux density calculation was in the researchers’ attention for a very long time. Important results in the domain, concerning the air-gap magnetic field in synchronous and respectively induction machine were published in the twenties of the last century, as were the works of Weber [1], Spooner [2], Carter [3] and Wieseman [4]. Heller’s book, [5], represents a synthesis of Heller entire work, containing also the most important other contributions published until the sixties last century concerning the air-gap field in the induction machine. In the last years, two types of electric machines were in the attention of the researchers, the synchronous permanent magnet (PM) machine with PMs in the air-gap [6–8], and the switched reluctance machine (SRM) [9,10]. In the case of rotating or linear transverse flux reluctance motors, the air-gap magnetic field calculation is important too and some results are given in [11,12], but the problem is not too different from the case of SRM’s. The calculation of the air-gap permeance of the double-slotted electric machines was also done, results being published, as in [13–15]. Since the case of the machines with PMs on the surface of rotor and stator was quite extensively studied recently, [6–8] for instance, in this paper will be considered

260

I.-A. Viorel et al. / Air-Gap Magnetic Field of the Unsaturated Slotted Electric Machines

only the case when there are no PMs in the air-gap, or buried inside the stator or rotor core. The stator and rotor core surfaces are considered as equipotential ones; therefore the magnetic field in the air-gap can be analytically described by a one-dimension (1D) model. The air-gap flux density variation versus circumferential coordinate, calculated via 2D-FEM analysis, is, as expected, continuous and smooth, without local points where its first derivative became zero on the entirely domain, half of the slot pitch, but at the domain extremities. Due to this important observation, the air-gap flux density variation requires a simple analytical estimating function in the cases when only Carter’s factor, torques or back emf have to be calculated. When main phase inductance depends on the rotor position, SRM for instance, the air-gap flux density variation must be approximated by an analytical function which gives values as closed as possible to the actual ones, since a simple estimating function might introduce errors. Different analytical estimations of the air-gap flux density variation versus circumferential coordinate are considered in the paper. The obtained results are compared between them and with the 2D-FEM calculated values on a simplified machine model. The Carter’s factor, which represents a criterion for the estimation accuracy, is calculated too and compared with that obtained via 2D-FEM analysis. The best fitted estimations for different cases are discussed and pertinent conclusions regarding the analytical estimations of the air-gap flux density are presented.

2. 2D-FEM Analysis The 2D-FEM analysis was carried on a simple model. The model’s structure contains two slots, a coil on the stator and a non-salient rotor. For symmetry reasons the stator length is equal to two tooth pitches. The notations contain the suffixes S, R for stator and rotor. The topology was parameterized in a way to make possible adequate variation of the most important dimensions, only the tooth pitch, which is the circumferential length, was kept constant. A linear layout was considered, but it does not affect the generality. In the air-gap were considered six layers in order to obtain a more homogenous distribution of the vector magnetic potential affected by the magnetic permeability difference between the core and the air-gap domain. The flux density, calculated in the middle of the air-gap, has a smooth and continuous variation, Figs 2 and 3. If the ratio between the tooth width and the double of the stator yoke is small, then the iron-core is unsaturated even if the air-gap flux density has large values, Table 1, via 2D-FEM analysis, when t/g = 37.5, wt/2hyS = 0.667, wt/2hyR = 0.333. Based on 2D-FEM analysis obtained values, the Carter’s factor is calculated as the ratio between the peak and the average value of the air-gap flux density, K C = Bg max / Bgav

(1)

The saturation coefficient Ksat results as: K sat = Bg max unsat / Bg max

(2)


261

Table 1. Saturation Ksat and Carter KC factors Bgmax Ksat Kc

0.8546 1.011 1.3597

1.16 1.015 1.362

1.433 1.042 1.368

1.526 1.183 1.372

Figure 1. Machine model with nonsalient rotor.

where the saturated peak air-gap flux density value Bgmax is obtained in the tooth axis via 2D-FEM analysis and the unsaturated air-gap flux density value is: Bg max unsat = μ 0 F / 2 g

(3)

with the following usual notations, Fig. 1, t – tooth pitch, wt – tooth width, ws – slot width, hyS, hyR – stator and rotor yoke radial length, g – air-gap radial length, F – coil mmf in ampere turns, µ0 – air-gap permeability (4π10–7H/m). The air-gap flux density average value Bgav is given by: Bgav =

2 t

t/2

∫B

g

( x)dx

(4)

0

where Bg(x) are the air-gap flux density values obtained at equidistant points via 2D-FEM analysis, the integral being numerically calculated. Examples of the air-gap flux density variation versus the circumferential coordinate x ∈ [0,t/2], calculated by FEM, are given for two values of mmf in each case, in Figs 2 and 3. The values of the actual mmf, the pole pitch to air-gap length ratio t/g, the slot width to double air-gap length ws/2g ratio, peak and average flux density Bgmax, Bgav, Carter and saturation factors KC, Ksat for all cases presented in Figs 2 and 3 are shown in Table 2. Two important remarks should be made considering the curves shown in Figs 2 and 3, and the values given in Table 2:

262


Table 2. Representative values for the curves given in Figs 2 and 3 Figure

t/g

ws/2g

Fig. 2

37.5

6.25

Fig. 3

75

25

F[A] 2200 3800 1000 1800

Figure 2. Flux density variation, t/g = 37.5, ws/2g = 6.25.

Bgmax[T] 0.855 1.433 0.776 1.383

Bgav[T] 0.628 1.047 0.304 0.539

KC 1.36 1.368 2.556 2.568

Ksat 1.011 1.042 1.011 1.022

Figure 3. Flux density variation, t/g = 75, ws/2g = 25.

i) The air-gap flux density variation versus circumferential coordinate is represented by smooth and continuous curves. ii) Even at important peak flux density values, the saturation is not important for the case considered, wt/2h y < 1 and consequently the Carter’s factors are not dependent on the mmf. 3. Air-Gap Flux Density Variation One of the analytical approximation of the air-gap flux density variation quite intensively employed is based on the air-gap variable equivalent permenace, a method which was first introduced in the case of induction motor [5,15] and was later extended to other motors, SRM [16] or transverse flux motor (TFM) [12], for example. The air-gap variable equivalent permeance can be easily defined and allows the calculation of the air-gap magnetic field in the case of double slotted machine. Its accuracy depends on the accuracy of the estimation for the air-gap flux density variation over a tooth pitch. If Bgmin is the minimum value of the air-gap flux density, then: Bg ( x) ≅ Bgav + 0.5( Bg max − Bg min ) cos

2x π, t

x ∈ [0, t / 2]

(5)

The variable equivalent air-gap permeance is defined as: P( x) =

ΔBg 1 Bg ( x) ⎛ Bgav 2x ⎞ 1 =⎜ + cos π ⎟ ⎜ g Bg max ⎝ Bg max Bg max t ⎟⎠ g

(6)


263

where the air-gap topology coefficient β is [6]: Bg max − Bg min

β=

2 Bg max

=

ΔBg Bg max

⎛ 1 β = 0.5 ⎜1 − ⎜ 1 + ( ws / 2 g )2 ⎝

⎞ ⎟ ⎟ ⎠

(7)

The air-gap variable equivalent permenace, function of the circumferential coordinate x, comes: P( x) =

1 1 ⎛ 2x ⎞ ⎜ 1 + pr cos π ⎟ g KC ⎝ t ⎠

(8)

where the permeance coefficient pr = KCβ or, as proposed in [15], pr =

γ =

⎛γ gπ ⎞ 4 β K C sin ⎜ ⎟ π ⎝β t 2⎠

(9)

2 ⎡ ⎛w ⎞ ⎤ w 4 ⎢ ws a tan( s ) − ln 1 + ⎜ s ⎟ ⎥ 2g π ⎢ 2g ⎝ 2 g ⎠ ⎥⎦ ⎣

(10)

Some equation, eventually more accurate than (5), can be developed, such as: i) The one proposed by Weber [1] t − ws ⎛x⎞ Bg ( x) = Bg max (1 − 2β sin 2 a ⎜ ⎟ π ) , a = , ws ⎝t⎠

x ∈ [0, t / 2]

(11)

ii) The one proposed by Heller [5] Bg ( x) = Bg max, ,

x ∈ [0, β wt )

Bg ( x) = Bg max (1 − β + β cos y ) , y=

x ∈ [ β wt , t / 2] ,

πβ wt π x− t / 2 − β wt t / 2 − β wt

(12)

iii) A nonsinusoidal variation, considering a simplified flux lines topology in the air-gap, [7] Bg ( x) = Bg max, ,

x ∈ [0, wt / 2] −1

⎡ ⎤ π Bg ( x) = Bg max, ⎢1 + ( x − wt / 2) ⎥ , 2 g ⎣ ⎦

x ∈ ( wt / 2, t / 2]

(13)

264


Figure 4. Air-gap flux density variation, referred to its peak value, FEM and proposed approximations, t/g = 37.5, ws/2g = 6.25, F = 3800 A.

Figure 5. Air-gap flux density variation, referred to its peak value, FEM and proposed approximations, t/g = 75, ws/2g = 25, F = 1000 A.

iv) An exponential approximation, considering x = 0 in a slot axis, obtained through a curve fitting procedure, given by: Bg ( x) = Bg max

1 , 1 + 10( β ws − x )

x ∈ [0, t / 2]

(14)

A comparison between the FEM analysis and different proposed approximations of the air-gap flux density referred to its peak value is given in Figs 4 and 5. In Fig. 4 the calculated values obtained in the case of t/g = 37.5, ws/2g = 6.25 and F = 3800 AT, as in Fig. 2 and in Fig. 5 in the case of t/g = 75, ws/2g = 25 and F = 1000 AT as in Fig. 3, are shown. From all the cases considered were chosen these two since there is an important difference between tooth pitch to air-gap length and respectively slot width to double air-gap length ratios, which allows for a large degree of generality of the results and conclusions. 4. Carter’s Factor Basically, the Carter’s factor is defined by (1), the average air-gap flux density being calculated accordingly to (4). Only the air-gap flux density variation on a half of a tooth pitch is considered due to the symmetry against the tooth axis. Carter’s factor can be calculated using different equations such as, [8]: kC 1

⎡ ⎤ 1 = ⎢1 − ⎥ ⎢⎣ t / ws ( 5 g / ws + 1) ⎥⎦

kC 2

⎡ 2w = ⎢1 − s tπ ⎢ ⎣

−1

2 ⎛ ws g ⎛ 1 ⎛ ws ⎞ ⎞ ⎞ ⎥⎤ ⎜ a tan( ) − ln ⎜1 + ⎜ ⎟ ⎟ ⎟ ⎜ 2 g ws ⎜⎝ 4 ⎝ g ⎠ ⎟⎠ ⎟ ⎥ ⎝ ⎠⎦

(15)

−1

(16)


265

Table 3. Carter’s factor values, t/g = 150 ws/2g KC FEM (15) (16) (17)

25

31.25

37.5

43.75

50

1.446 1.435 1.424 1.433

1.637 1.628 1.61 1.622

1.886 1.882 1.854 1.87

2.225 2.231 2.187 2.209

2.718 2.739 2.666 2.669

Table 4. Carter’s factor values, t/g = 37.5 ws/2g KC FEM (15) (16) (17)

6.25

7.813

9.375

10.938

12.5

1.359 1.313 1.311 1.338

1.515 1.461 1.455 1.49

1.714 1.652 1.639 1.685

1.978 1.904 1.88 1.94

2.341 2.25 2.207 2.293

⎡ w 4 g ⎛ π ws ⎞ ⎤ kC 3 = ⎢1 − s + ln ⎜1 + ⎟⎥ t tπ ⎝ 4 g ⎠⎦ ⎣

−1

(17)

In Table 3 and 4 the calculated values of Carter’s factor for unsaturated machine, Ksat

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